- What does “percent” mean? - First write the percent as a fraction with denominator \(100\). - Simplify the fraction as much as possible.

1. Write the percent as a fraction with denominator \(100\): \(12\% = \frac{12}{100}\). 2. Simplify the fraction: \(\frac{12}{100} = \frac{3}{25}\) by dividing the numerator and denominator by \(4\).

c) \(\frac{3}{25}\)

- Can you write the fraction with denominator \(100\)? - What does the numerator mean when the denominator is \(100\)?

1. Multiply the numerator and denominator by \(5\): \(\frac{7}{20} = \frac{35}{100}\). 2. The fraction \(\frac{35}{100}\) is equivalent to \(35\%\).

\(35\%\)

- How many equal parts are in the strip? - How many parts are shaded? - Write the shaded portion as a fraction first. - How do you convert fourths to percents?

1. The strip has \(4\) equal parts. 2. Three parts are shaded. 3. The shaded portion is \(\frac{3}{4}\). 4. Convert to a percent: \(\frac{3}{4} = \frac{75}{100} = 75\%\).

\(75\%\) of the strip is shaded.

- Count all the equal sectors in the circle. - How many sectors are shaded? - Write the fraction with denominator \(100\).

1. The circle is divided into \(5\) equal sectors. 2. Two sectors are shaded. 3. The shaded portion is \(\frac{2}{5}\). 4. Convert to a percent: \(\frac{2}{5} = \frac{40}{100} = 40\%\).

\(40\%\) of the circle is shaded.

- Each grid is one whole divided into \(100\) equal cells. - Interpret the percent as a number out of \(100\). - Count the shaded cells in each candidate model and match that count to the stated percent.

1. Each panel contains \(100\) equal cells. 2. Panel a) has \(65\) shaded cells, panel b) has \(35\) shaded cells, and panel c) has \(30\) shaded cells. 3. A percent is a rate per \(100\), so \(35\%\) means \(35\) shaded cells out of \(100\). 4. Therefore panel b) represents \(35\%\).

Panel b), because it has \(35\) shaded cells out of \(100\).

- Think about what a percent compares with the full invited group. - Would the fraction of students attending be the same in parts a) and b)?

1. All \(100\) out of \(100\) students is the whole group, or \(100\%\). 2. No students out of \(100\) is \(0\%\).

a) \(100\%\) b) \(0\%\)

- Count all the little squares in the grid. - Count the shaded squares. - Percent means a count for every one hundred.

1. The grid has \(10\times10=100\) squares. 2. Ten of them are shaded, so the shaded share is \(\frac{10}{100}=10\%\).

\(10\%\); the grid has \(100\) small squares, and \(10\) are shaded.

- Four tenths is how many hundredths? - What does the word “percent” mean?

1. \(0.4=\frac{4}{10}=\frac{40}{100}\). 2. Forty out of a hundred is \(40\%\).

\(40\%\), because \(0.4=\frac{40}{100}\).

- What part of a group is \(50\%\)? - How many equal parts make one whole group? - Think about the half that is not mentioned.

1. Fifty percent is one half of the club. 2. The other half has the same number of students, so the total is \(14\times2=28\).

\(28\) students; the \(14\) students are half the club, so another equal group of \(14\) makes the whole.

- What does “percent” mean? - Write each percent as a fraction with denominator \(100\). - What common factors do the numerator and denominator have?

1. For \(12\%\): \(\frac{12}{100} = \frac{3}{25}\) after dividing by \(4\). 2. For \(45\%\): \(\frac{45}{100} = \frac{9}{20}\) after dividing by \(5\). 3. For \(125\%\): \(\frac{125}{100} = \frac{5}{4}\) after dividing by \(25\). 4. For \(8\%\): \(\frac{8}{100} = \frac{2}{25}\) after dividing by \(4\).

\(12\% = \frac{3}{25}\) \(45\% = \frac{9}{20}\) \(125\% = \frac{5}{4}\) \(8\% = \frac{2}{25}\)

- What can you multiply each denominator by to get \(100\)? - It may help to simplify a fraction before writing an equivalent fraction with denominator \(100\). - Percent means “per hundred.”

1. Multiply the numerator and denominator of \(\frac{3}{4}\) by \(25\): \(\frac{3}{4} = \frac{75}{100} = 75\%\). 2. Simplify \(\frac{18}{60}\) by dividing by \(6\), then multiply the numerator and denominator by \(10\): \(\frac{18}{60} = \frac{3}{10} = \frac{30}{100} = 30\%\). 3. Multiply the numerator and denominator of \(\frac{11}{20}\) by \(5\): \(\frac{11}{20} = \frac{55}{100} = 55\%\). 4. Compare the percents: \(30\% < 55\% < 75\%\).

\(\frac{3}{4} = \frac{75}{100} = 75\%\) \(\frac{18}{60} = \frac{30}{100} = 30\%\) \(\frac{11}{20} = \frac{55}{100} = 55\%\) Order: \(30\% < 55\% < 75\%\)

- How can the greatest common factor help you simplify each fraction? - How can you write three fourths or nine tenths as hundredths? - You can compare the fractions by writing them as decimals or percents.

1. The greatest common factor of \(48\) and \(64\) is \(16\). Thus, \(\frac{48}{64} = \frac{3}{4} = 0.75 = 75\%\). 2. The greatest common factor of \(126\) and \(140\) is \(14\). Thus, \(\frac{126}{140} = \frac{9}{10} = 0.9 = 90\%\). 3. The greatest common factor of \(81\) and \(108\) is \(27\). Thus, \(\frac{81}{108} = \frac{3}{4} = 0.75 = 75\%\). 4. Compare the percents: \(75\% = 75\% < 90\%\).

\(\frac{48}{64} = 75\%\) \(\frac{126}{140} = 90\%\) \(\frac{81}{108} = 75\%\) Order: \(\frac{48}{64} = \frac{81}{108} < \frac{126}{140}\), or \(75\% = 75\% < 90\%\)

- What can you multiply each denominator by to get \(100\)? - Multiply the numerator by the same number. - A fraction with denominator \(100\) can be written directly as a percent.

1. Multiply the numerator and denominator of \(\frac{3}{4}\) by \(25\): \(\frac{3}{4} = \frac{75}{100} = 75\%\). 2. Multiply the numerator and denominator of \(\frac{7}{10}\) by \(10\): \(\frac{7}{10} = \frac{70}{100} = 70\%\). 3. Multiply the numerator and denominator of \(\frac{13}{20}\) by \(5\): \(\frac{13}{20} = \frac{65}{100} = 65\%\). 4. Multiply the numerator and denominator of \(\frac{18}{50}\) by \(2\): \(\frac{18}{50} = \frac{36}{100} = 36\%\). 5. Multiply the numerator and denominator of \(\frac{1}{5}\) by \(20\): \(\frac{1}{5} = \frac{20}{100} = 20\%\).

a) \(75\%\) b) \(70\%\) c) \(65\%\) d) \(36\%\) e) \(20\%\)

- Can you simplify the fraction first? - When the denominator is greater than \(100\), can you divide it to get \(100\)? - Think about the decimal value of eighths.

1. Simplify \(\frac{21}{70}\) by dividing by \(7\): \(\frac{21}{70} = \frac{3}{10} = \frac{30}{100} = 30\%\). 2. Simplify \(\frac{12}{16}\) by dividing by \(4\): \(\frac{12}{16} = \frac{3}{4} = \frac{75}{100} = 75\%\). 3. Divide to convert \(\frac{3}{8}\) to a decimal: \(3 \div 8 = 0.375\). Therefore, \(\frac{3}{8} = 37.5\%\). 4. Divide the numerator and denominator of \(\frac{24}{400}\) by \(4\): \(\frac{24}{400} = \frac{6}{100} = 6\%\).

a) \(30\%\) b) \(75\%\) c) \(37.5\%\) d) \(6\%\)

- What denominator does the word “percent” represent? - How can you write a decimal with one decimal place as a fraction in tenths? - Multiply a fraction's numerator and denominator by the same number to make an equivalent fraction.

1. Write \(A = 0.8\) as hundredths: \(0.8 = \frac{8}{10} = \frac{80}{100}\). 2. Write \(B = \frac{4}{5}\) as hundredths: \(\frac{4}{5} = \frac{80}{100}\). 3. Write \(C = 8\%\) as hundredths: \(8\% = \frac{8}{100}\). 4. Write \(D = \frac{80}{10}\) as hundredths: \(\frac{80}{10} = \frac{800}{100}\). 5. Values \(A\) and \(B\) are equal because both are \(\frac{80}{100}\). 6. Value \(A\) is \(80\%\), while \(C\) is \(8\%\). Therefore, \(A\) is \(10\) times \(C\).

a) \(A\) and \(B\) are equal because \(0.8 = \frac{80}{100}\) and \(\frac{4}{5} = \frac{80}{100}\). b) \(A = 80\%\), while \(C = 8\%\). Therefore, \(A\) is \(10\) times \(C\).

- What does “percent” mean? - First write each value as a fraction with denominator \(100\). - Is there a number that divides both the numerator and denominator?

1. Write each percent as a fraction with denominator \(100\): \(20\% = \frac{20}{100}\), \(25\% = \frac{25}{100}\), \(15\% = \frac{15}{100}\), and \(60\% = \frac{60}{100}\). 2. Simplify each fraction: a) \(\frac{20}{100} = \frac{1}{5}\) by dividing by \(20\). b) \(\frac{25}{100} = \frac{1}{4}\) by dividing by \(25\). c) \(\frac{15}{100} = \frac{3}{20}\) by dividing by \(5\). d) \(\frac{60}{100} = \frac{3}{5}\) by dividing by \(20\).

a) \(\frac{1}{5}\) b) \(\frac{1}{4}\) c) \(\frac{3}{20}\) d) \(\frac{3}{5}\)

- How can you change a denominator of \(5\) or \(20\) to \(100\)? - When converting a percent to a fraction with a given denominator, first write the percent as a fraction over \(100\), then simplify.

1. For a), write the fraction with denominator \(100\): \(\frac{2}{5} = \frac{40}{100} = 40\%\). 2. For b), write the percent as a fraction and simplify: \(18\% = \frac{18}{100} = \frac{9}{50}\). The missing number is \(9\). 3. For c), write the fraction with denominator \(100\): \(\frac{7}{20} = \frac{35}{100} = 35\%\). 4. For d), write the percent as a fraction and simplify: \(88\% = \frac{88}{100} = \frac{22}{25}\). The missing number is \(22\).

a) \(40\%\) b) \(9\) c) \(35\%\) d) \(22\)

- What does “percent” mean? - Can you write each fraction with denominator \(100\)? - What happens to the decimal point when you multiply a decimal by \(100\)?

1. \(\frac{3}{10} = \frac{30}{100} = 30\%\). 2. \(0.15 = \frac{15}{100} = 15\%\). 3. \(\frac{11}{50} = \frac{22}{100} = 22\%\). 4. \(0.085 \times 100 = 8.5\), so \(0.085 = 8.5\%\). 5. \(\frac{1}{4} = \frac{25}{100} = 25\%\).

a) \(30\%\) b) \(15\%\) c) \(22\%\) d) \(8.5\%\) e) \(25\%\)

- What percent represents one half? - Can you write each fraction with denominator \(100\)? - Add the two fruit-juice percents.

1. Apple juice: \(\frac{7}{20} = \frac{35}{100} = 35\%\). 2. Cherry juice: \(\frac{11}{50} = \frac{22}{100} = 22\%\). 3. Add the fruit-juice portions: \(35\% + 22\% = 57\%\). 4. Since half is \(50\%\) and \(57\% > 50\%\), more than half of the drink is fruit juice.

a) Apple juice: \(35\%\); cherry juice: \(22\%\) b) Yes. The total fruit-juice portion is \(57\%\), which is greater than \(50\%\).

- Can you write each fraction with denominator \(100\)? - How many places does the decimal point move when you convert a decimal to a percent? - What does “percent” mean?

1. For a), \(\frac{7}{25} = \frac{28}{100} = 28\%\). 2. For b), \(\frac{13}{200} = 0.065 = 6.5\%\). 3. For c), \(0.045 \times 100 = 4.5\), so \(0.045 = 4.5\%\). 4. For d), \(1\frac{2}{5} = 1.4 = 140\%\). 5. For e), \(3 \div 8 = 0.375\), so \(\frac{3}{8} = 37.5\%\).

a) \(28\%\) b) \(6.5\%\) c) \(4.5\%\) d) \(140\%\) e) \(37.5\%\)

- Can you write each fraction with denominator \(100\)? - What does “percent” mean? - Once the denominators are the same, which numerator is greatest?

1. Soccer: \(\frac{1}{4} = \frac{25}{100} = 25\%\). 2. Choir: \(\frac{3}{20} = \frac{15}{100} = 15\%\). 3. Drama: \(\frac{2}{25} = \frac{8}{100} = 8\%\). 4. Woodworking: \(\frac{7}{50} = \frac{14}{100} = 14\%\). 5. Since \(25\% > 15\% > 14\% > 8\%\), soccer is the most popular of the four clubs.

Soccer: \(25\%\); choir: \(15\%\); drama: \(8\%\); woodworking: \(14\%\). Soccer is the most popular of these clubs.

- What does \(100\%\) represent in this situation? - What percent of the foods is already accounted for? - How can you find the remaining percent?

1. Add the two nonvegan portions: \(45\% + 35\% = 80\%\). 2. Subtract from the whole: \(100\% - 80\% = 20\%\).

\(20\%\) of the foods are vegan.

- How many squares are in the grid altogether? - Convert each decimal, fraction, or percent to a number of squares. - How do you simplify the fraction for the white squares?

1. The grid has \(10 \times 10 = 100\) squares. 2. Blue squares: \(0.3 = 30\%\), so \(30\) squares are blue. 3. Red squares: \(\frac{15}{100}\) of \(100\) is \(15\) squares. 4. Yellow squares: \(20\%\) of \(100\) is \(20\) squares. 5. The total number of colored squares is \(30 + 15 + 20 = 65\). 6. The number of white squares is \(100 - 65 = 35\). The white portion is \(\frac{35}{100} = \frac{7}{20}\).

\(35\) squares are white. The white portion is \(\frac{7}{20}\).

- Percent means “per hundred.” - Multiply a decimal by \(100\) to write it as a percent. - Rewrite a fraction with a denominator of \(100\) when that is convenient.

1. \(\frac{2}{5} = 0.4 = 40\%\). 2. \(0.08 = \frac{8}{100} = \frac{2}{25} = 8\%\). 3. \(15\% = 0.15 = \frac{15}{100} = \frac{3}{20}\). 4. \(0.004 = \frac{4}{1000} = \frac{1}{250} = 0.4\%\). 5. Complete the table. <table> <tr><th>Fraction</th><th>Decimal</th><th>Percent</th></tr> <tr><td>\(\frac{2}{5}\)</td><td>\(0.4\)</td><td>\(40\%\)</td></tr> <tr><td>\(\frac{2}{25}\)</td><td>\(0.08\)</td><td>\(8\%\)</td></tr> <tr><td>\(\frac{3}{20}\)</td><td>\(0.15\)</td><td>\(15\%\)</td></tr> <tr><td>\(\frac{1}{250}\)</td><td>\(0.004\)</td><td>\(0.4\%\)</td></tr> </table>

<table> <tr><th>Fraction</th><th>Decimal</th><th>Percent</th></tr> <tr><td>\(\frac{2}{5}\)</td><td>\(0.4\)</td><td>\(40\%\)</td></tr> <tr><td>\(\frac{2}{25}\)</td><td>\(0.08\)</td><td>\(8\%\)</td></tr> <tr><td>\(\frac{3}{20}\)</td><td>\(0.15\)</td><td>\(15\%\)</td></tr> <tr><td>\(\frac{1}{250}\)</td><td>\(0.004\)</td><td>\(0.4\%\)</td></tr> </table>

- Think about how a fraction and a decimal can represent the same value. - Recall what “percent” means in relation to \(100\). - Use the next digit beyond the tenths place of the percent to decide how to round.

1. For a), \(4 \div 9 = 0.444\ldots\), so \(\frac{4}{9} = 44.444\ldots\% \approx 44.4\%\). 2. For b), \(0.0725 \times 100 = 7.25\), so \(0.0725 = 7.25\% \approx 7.3\%\). 3. For c), \(11 \div 13 = 0.846153\ldots\), so \(\frac{11}{13} = 84.6153\ldots\% \approx 84.6\%\). 4. For d), \(0.009 \times 100 = 0.9\), so \(0.009 = 0.9\%\).

a) \(44.4\%\) b) \(7.3\%\) c) \(84.6\%\) d) \(0.9\%\)

- A full circle measures \(360^\circ\). - Five equal sectors divide both \(360^\circ\) and \(100\%\) equally. - Dividing one sector in half also divides its angle and percent in half.

1. One of five equal sectors has central angle \(360^\circ\div5=72^\circ\). 2. One sector represents \(100\%\div5=20\%\) of the whole. 3. Each half of that sector has central angle \(72^\circ\div2=36^\circ\). 4. Each smaller sector represents \(20\%\div2=10\%\) of the whole.

a) \(72^\circ\) and \(20\%\) b) Each smaller sector is \(36^\circ\) and represents \(10\%\).

- First identify the whole, the part, and the percent in each situation. - Can you rewrite each situation in the form “\(p\%\) of the whole is the part”? - For b), write the part-to-whole ratio as a fraction and find an equivalent fraction with denominator \(100\). - For c), you know the amount represented by \(10\%\). How many groups of \(10\%\) make \(100\%\)?

1. In part a), the part is unknown. Compute \(400\,\text{m} \times 0.15 = 60\,\text{m}\). 2. In part b), the percent is unknown. Compute \(\frac{12}{60} = \frac{1}{5} = \frac{20}{100} = 20\%\). 3. In part c), the whole is unknown. Compute \(\$5 \div 0.10 = \$50\).

a) Part; \(60\,\text{m}\) b) Percent; \(20\%\) c) Whole; \(\$50\)

- Decide what quantity represents \(100\%\). - Use the known part and its percent to find the whole. - Check by finding \(40\%\) of your result.

1. The whole is the total planned route. 2. Since \(12\,\text{km}\) is \(40\%\) of the route, the total is \(12\,\text{km} \div 0.40 = 30\,\text{km}\). 3. Check: \(30\,\text{km} \times 0.40 = 12\,\text{km}\), which matches the given distance.

The whole is the total planned route, and its length is \(30\,\text{km}\). The check gives \(30\,\text{km} \times 0.40 = 12\,\text{km}\).

- First write each described portion as a fraction. - What does “percent” mean? - How can you write each fraction with denominator \(100\)?

1. “One out of every two” means \(\frac{1}{2}\): \(\frac{1}{2} = \frac{50}{100} = 50\%\). 2. “Three out of four” means \(\frac{3}{4}\): \(\frac{3}{4} = \frac{75}{100} = 75\%\). 3. “One fifth” means \(\frac{1}{5}\): \(\frac{1}{5} = \frac{20}{100} = 20\%\).

a) \(50\%\) b) \(75\%\) c) \(20\%\)

- Write each percent as a decimal or fraction. - Divide the part by that decimal to find the whole. - What does \(100\%\) mean for the relationship between the part and the whole?

1. Divide each given part by the percent written as a decimal. 2. a) \(G = 7 \div 0.10 = 70\) 3. b) \(G = 18 \div 0.50 = 36\) 4. c) \(G = 11 \div 0.25 = 44\) 5. d) \(G = 34 \div 1 = 34\)

a) \(G = 70\) b) \(G = 36\) c) \(G = 44\) d) \(G = 34\)

- Find how many items correspond to \(1\%\), or divide by the percent written as a decimal. - The original inventory represents \(100\%\).

1. a) \(12 \div 0.15 = 80\) bracelets 2. b) \(45 \div 0.60 = 75\) key chains 3. c) \(21 \div 0.30 = 70\) postcards

a) \(80\) bracelets b) \(75\) key chains c) \(70\) postcards

- You may first find \(1\%\), or divide the part by the percent written as a decimal. - For part d), decide whether the whole should be greater or less than \(330\) when \(330\) represents \(150\%\).

1. a) \(G = 42 \div 0.12 = 350\) 2. b) \(G = 105 \div 0.35 = 300\) 3. c) \(G = 144 \div 0.80 = 180\) 4. d) \(G = 330 \div 1.50 = 220\)

a) \(G = 350\) b) \(G = 300\) c) \(G = 180\) d) \(G = 220\)

- What percent represents the whole class? - First write all the given portions as percents. - Subtract the known total from the whole.

1. Convert the known portions: \(\frac{1}{4} = 25\%\) for apple juice, and \(0.3 = 30\%\) for water. 2. Add the known percents: \(25\% + 30\% + 35\% = 90\%\). 3. Subtract from the whole: \(100\% - 90\% = 10\%\).

a) Apple juice: \(25\%\); water: \(30\%\) b) \(10\%\) of the class chose lemonade.

- Think about whether each fraction can represent an exact whole number of hundredths. - Simplifying first may make the relationship to \(100\) easier to see. - For any fraction that does not give a whole-number percent, explain what prevents an exact whole number of hundredths.

1. A fraction is a whole-number percent when \(100\) times its numerator is divisible by its denominator. 2. For a), \(\frac{3}{5} \times 100 = 60\), so \(\frac{3}{5} = 60\%\). 3. For b), \(\frac{5}{8} \times 100 = \frac{500}{8} = 62.5\), so the percent is not a whole number. 4. For c), \(\frac{9}{20} \times 100 = 45\), so \(\frac{9}{20} = 45\%\). 5. For d), \(\frac{7}{30} \times 100 = \frac{70}{3} = 23.333\ldots\), so the percent is not a whole number.

Whole-number percents: a) \(\frac{3}{5} = 60\%\) c) \(\frac{9}{20} = 45\%\) Not whole-number percents: b) \(\frac{5}{8} = 62.5\%\), because \(500\) is not divisible by \(8\) d) \(\frac{7}{30} = 23.333\ldots\%\), because \(700\) is not divisible by \(30\)

- The full battery represents \(100\%\). - Use \(210\,\text{Wh}\) and \(35\%\) to find the whole capacity.

1. The \(210\,\text{Wh}\) represents \(35\%\) of the full capacity. 2. Divide by the percent written as a decimal: \(210\,\text{Wh} \div 0.35 = 600\,\text{Wh}\).

The battery’s full capacity is \(600\,\text{Wh}\).

- Decide which quantity is the part and which is the whole. - How many groups of \(20\%\) make \(100\%\)?

1. The \(120\,\text{g}\) of hazelnuts represents \(20\%\) of the total. 2. Find the whole: \(120\,\text{g} \div 0.20 = 600\,\text{g}\).

The trail mix has a total mass of \(600\,\text{g}\).

- Can you write each fraction with denominator \(10\), \(100\), or \(1000\)? - What does “percent” mean? - How does the decimal point move when you multiply a decimal by \(100\)?

1. Soccer: \(\frac{2}{5} = 0.4 = 40\%\). 2. Swimming: \(\frac{3}{20} = 0.15 = 15\%\). 3. Basketball: \(\frac{1}{4} = 0.25 = 25\%\). 4. Gymnastics: \(\frac{1}{10} = 0.1 = 10\%\).

1. \(0.4 = 40\%\) 2. \(0.15 = 15\%\) 3. \(0.25 = 25\%\) 4. \(0.1 = 10\%\)

- Write every term in the same form. - How many hundredths are in three tenths? - Why is the sum easier to find when all terms are percents?

1. Convert \(\frac{3}{10}\) to a percent: \(\frac{3}{10} = \frac{30}{100} = 30\%\). 2. Convert \(0.13\) to a percent: \(0.13 = 13\%\). 3. Add the three percent values: \(30\% + 12\% + 13\% = 55\%\).

\(55\%\)

- Write each part as the number in the group divided by the total number of students. - Since \(25 \times 4 = 100\), fractions with denominator \(25\) can be rewritten with denominator \(100\). - The percentages for all three groups must add to \(100\%\).

1. For soccer, the fraction is \(\frac{15}{25} = \frac{3}{5}\). Since \(\frac{3}{5} = \frac{60}{100}\), the percent is \(60\%\). 2. For tennis, the fraction is \(\frac{6}{25}\). Dividing gives \(6 \div 25 = 0.24\), so the percent is \(24\%\). 3. The number who do not play a sport is \(25 - 15 - 6 = 4\). The fraction is \(\frac{4}{25} = \frac{16}{100}\), so the percent is \(16\%\).

a) \(\frac{3}{5}\) and \(60\%\) b) \(0.24\) and \(24\%\) c) \(16\%\)

- Rewrite every value in the same form, such as all decimals or all percents. - To convert a fraction to a decimal, divide the numerator by the denominator. - A percent means a number out of \(100\). - Simplifying fractions may make equivalent values easier to recognize.

1. Rewrite each value as a decimal or percent: \(\frac{3}{4} = 0.75 = 75\%\) \(0.75 = 75\%\) \(7.5\% = 0.075\) \(75\% = 0.75\) \(\frac{75}{10} = 7.5 = 750\%\) \(\frac{6}{8} = \frac{3}{4} = 0.75 = 75\%\) 2. Therefore, \(\frac{3}{4}\), \(0.75\), \(75\%\), and \(\frac{6}{8}\) are equivalent. 3. The values \(7.5\%\) and \(\frac{75}{10}\) are not equivalent to any other value in the list.

Equivalent group: \(\frac{3}{4}\), \(0.75\), \(75\%\), and \(\frac{6}{8}\). The values \(7.5\%\) and \(\frac{75}{10}\) each form a group by themselves.

- Read the gymnastics percent from the graph. - Use that known part to find \(100\%\) of the survey group.

1. Gymnastics represents \(15\%\) of the survey responses. 2. Since \(45\) is \(15\%\) of the group, the total is \(45 \div 0.15 = 300\) students.

\(300\) students participated in the survey.

- Count the rows and columns of small squares. - Work out how many of those squares are orange. - Write the part as a fraction before simplifying.

1. There are \(10\times10=100\) small squares. 2. Count \(28\) orange squares. Their share is \(\frac{28}{100}=28\%\). 3. Simplify \(\frac{28}{100}\) to \(\frac{7}{25}\).

a) \(28\%\) b) \(\frac{7}{25}\)

- Count the total number of small squares. - Count the blue squares and write the portion as a fraction. - Simplify the fraction, then convert it to a decimal and a percent. - The shaded and unshaded portions must add to \(1\), or \(100\%\).

1. The grid has \(4 \times 5 = 20\) squares. 2. Count the shaded squares: \(8\) squares are blue. 3. The shaded portion is \(\frac{8}{20} = \frac{2}{5} = 0.4 = 40\%\). 4. There are \(20 - 8 = 12\) unshaded squares. The unshaded portion is \(\frac{12}{20} = \frac{3}{5} = 0.6 = 60\%\).

a) Shaded portion: \(\frac{2}{5}\), \(0.4\), \(40\%\) b) Unshaded portion: \(\frac{3}{5}\), \(0.6\), \(60\%\)

- Count all the small tiles. - Count the gray tiles. - How can you write a fraction with denominator \(25\) as a fraction with denominator \(100\)? - Percent means “per hundred.”

1. The grid has \(5 \times 5 = 25\) tiles. 2. Count the gray tiles: \(6\) tiles are gray. 3. The gray portion is \(\frac{6}{25}\). 4. Multiply the numerator and denominator by \(4\): \(\frac{6}{25} = \frac{24}{100} = 24\%\).

The gray portion is \(\frac{6}{25}\), which is \(24\%\).

- Write \(25\%\) as a simple fraction. - Look for a sector that is one fourth of the circle. - What familiar angle measures one fourth of a full turn?

1. A portion of \(25\%\) is one fourth of a circle. The Homework sector represents \(25\%\). 2. One fourth of a circle has a central angle of \(90^\circ\). This right-angle sector is easy to recognize.

a) Homework represents \(25\%\). b) A portion of \(25\%\) is one fourth of a circle, which has a central angle of \(90^\circ\). This forms a right-angle sector.

- Notice whether each part asks about what remains or what was eaten. - How can you change a denominator of \(10\) to \(100\)?

1. There are \(10\) slices in all. Since \(3\) are shaded, \(10 - 3 = 7\) slices remain. The remaining portion is \(\frac{7}{10}\). 2. The eaten portion is \(\frac{3}{10}\). 3. Convert to a percent: \(\frac{3}{10} = \frac{30}{100} = 30\%\).

a) \(\frac{7}{10}\) of the pizza remains. b) \(30\%\) of the pizza has been eaten.

- How many spaces are there altogether? - How many shaded spaces are occupied? - How many unshaded spaces remain? - Percent means “per hundred.”

1. There are \(100\) parking spaces altogether. 2. The figure shows \(35\) shaded, occupied spaces. 3. The number of available spaces is \(100 - 35 = 65\). 4. Since there are \(100\) spaces in all, \(65\) available spaces represent \(65\%\).

\(65\%\) of the parking spaces are available.

- Find the shaded portion of the whole strip. - The whole represents \(100\%\).

1. The strip has \(10\) equal parts. 2. Nine parts are shaded, so the occupied portion is \(\frac{9}{10} = 90\%\). 3. The remaining portion is \(100\% - 90\% = 10\%\).

Occupied: \(90\%\) Available: \(10\%\)

- Count the total number of small squares in each grid. - Count the shaded squares. - Write each fraction with denominator \(100\).

1. In figure a), the grid has \(4 \times 5 = 20\) equal squares. Nine are shaded, so the shaded portion is \(\frac{9}{20} = \frac{45}{100} = 45\%\). 2. In figure b), the grid has \(2 \times 5 = 10\) equal squares. Three are shaded, so the shaded portion is \(\frac{3}{10} = \frac{30}{100} = 30\%\).

a) \(45\%\) b) \(30\%\)

- One full original amount represents \(100\%\). - Does \(1.25\) contain more or less than one full amount? - Write the decimal in hundredths.

1. One original amount is \(100\%\). 2. \(1.25=\frac{125}{100}\), so the new amount is \(125\%\) of the original. 3. It is more than the original amount.

\(125\%\), which is more than the original

- What does the percent sign say about the size of the comparison group? - How many groups of five fit into one hundred? - Compare the members in one hundred with Eva's claim.

1. There are \(100\div5=20\) groups of five visitors in one hundred. 2. One member in each group makes \(20\) members out of \(100\), or \(20\%\). 3. Eva has confused the group size with the number per hundred.

No. \(1\) out of \(5\) means \(20\) out of \(100\), or \(20\%\).

- Think about what share of a whole twenty percent represents. - If you know one share, how can you find all equal shares together?

1. Twenty percent is \(\frac{1}{5}\) of the club. 2. If one fifth is \(7\) members, five fifths is \(5\times7=35\) members.

\(35\) members, because \(20\%\) is one fifth and five groups of \(7\) make the whole club.

- Is the known number of tickets the whole or just part of the whole? - Can the given percent be expressed as a familiar fraction? - After recovering the total, how would you find the seats without tickets?

1. Seventy-five percent is three fourths. Three fourths of the seats are \(45\), so one fourth is \(45\div3=15\). 2. Four fourths make \(4\times15=60\) seats. 3. The empty seats number \(60-45=15\).

\(60\) seats in all; \(15\) empty seats

- How are fractions with denominator \(100\) related to decimals? - Can you write a fraction as an equivalent fraction with denominator \(100\)? - What happens to the decimal point when you convert a decimal to a percent?

1. First row: \(35\% = \frac{35}{100} = \frac{7}{20}\). As a decimal, \(35 \div 100 = 0.35\). 2. Second row: \(\frac{4}{5} = \frac{80}{100} = 80\%\). As a decimal, \(4 \div 5 = 0.8\). 3. Third row: \(0.12 = \frac{12}{100} = \frac{3}{25}\). As a percent, \(0.12 \times 100 = 12\%\). 4. The completed table is: <table><tr><th>Percent</th><th>Fraction (simplest form)</th><th>Decimal</th></tr><tr><td>\(35\%\)</td><td>\(\frac{7}{20}\)</td><td>\(0.35\)</td></tr><tr><td>\(80\%\)</td><td>\(\frac{4}{5}\)</td><td>\(0.8\)</td></tr><tr><td>\(12\%\)</td><td>\(\frac{3}{25}\)</td><td>\(0.12\)</td></tr></table>

<table><tr><th>Percent</th><th>Fraction (simplest form)</th><th>Decimal</th></tr><tr><td>\(35\%\)</td><td>\(\frac{7}{20}\)</td><td>\(0.35\)</td></tr><tr><td>\(80\%\)</td><td>\(\frac{4}{5}\)</td><td>\(0.8\)</td></tr><tr><td>\(12\%\)</td><td>\(\frac{3}{25}\)</td><td>\(0.12\)</td></tr></table>

- Can you write each percent as a fraction in simplest form? - Remember that a fraction bar also represents division. - What happens when you first write the percent as a fraction with denominator \(100\)?

1. For a), write \(15\%\) as a fraction and simplify: \(15\% = \frac{15}{100} = \frac{3}{20}\). Multiply the numerator and denominator by \(2\): \(\frac{3}{20} = \frac{6}{40}\). Therefore, the missing numerator is \(6\). 2. For b), write \(40\%\) as \(\frac{2}{5}\). Since \(\frac{12}{x} = \frac{2}{5}\), multiplying \(\frac{2}{5}\) by \(\frac{6}{6}\) gives \(\frac{12}{30}\), so \(x = 30\). 3. For c), simplify \(\frac{21}{70}\) to \(\frac{3}{10}\), then write it with denominator \(100\): \(\frac{3}{10} = \frac{30}{100} = 30\%\).

a) \(6\) b) \(30\) c) \(30\) and \(30\%\)

- Convert each value to a percent. - Find how far each percent is from \(50\%\). - The value with the smallest distance is closest to one-half.

1. Convert \(A\): \(\frac{14}{25}=\frac{56}{100}=56\%\). 2. Convert \(B\): \(0.49=49\%\). 3. Convert \(C\): \(\frac{123}{250}=0.492=49.2\%\). 4. The distances from \(50\%\) are \(6\) percentage points for \(A\), \(1\) percentage point for \(B\), and \(0.8\) percentage point for \(C\). 5. The smallest distance is \(0.8\) percentage point, so \(C\) is closest to one-half.

\(C=\frac{123}{250}\), because \(A=56\%\), \(B=49\%\), and \(C=49.2\%\), which is only \(0.8\) percentage point from \(50\%\).

- Work each conversion yourself before checking the stated result. - What common factor can you divide from each numerator and denominator? - Once a fraction has denominator \(100\), you can read the percent directly.

1. For \(\frac{14}{35}\), divide the numerator and denominator by their greatest common factor, \(7\): \(\frac{14}{35} = \frac{2}{5} = \frac{40}{100} = 40\%\). The result is correct. 2. For \(\frac{27}{45}\), divide the numerator and denominator by their greatest common factor, \(9\): \(\frac{27}{45} = \frac{3}{5} = \frac{60}{100} = 60\%\). The stated result, \(50\%\), is incorrect. 3. For \(\frac{66}{110}\), divide the numerator and denominator by their greatest common factor, \(22\): \(\frac{66}{110} = \frac{3}{5} = \frac{60}{100} = 60\%\). The result is correct.

1. Correct: \(\frac{14}{35} = \frac{2}{5} = 40\%\) 2. Incorrect: \(\frac{27}{45} = \frac{3}{5} = 60\%\), not \(50\%\) 3. Correct: \(\frac{66}{110} = \frac{3}{5} = 60\%\)

- Write each percent as a fraction in simplest form. - Can you rearrange each equation so that \(x\) is by itself? - For c), simplify the fraction before converting it to a percent.

1. For a), \(20\% = \frac{20}{100} = \frac{1}{5}\). Solve \(\frac{x}{125} = \frac{1}{5}\): \(x = 125 \times \frac{1}{5} = 25\). 2. For b), \(75\% = \frac{75}{100} = \frac{3}{4}\). Solve \(\frac{36}{x} = \frac{3}{4}\): \(3x = 36 \times 4 = 144\), so \(x = 144 \div 3 = 48\). 3. For c), simplify \(\frac{21}{28}\) by dividing by \(7\): \(\frac{21}{28} = \frac{3}{4} = 0.75 = 75\%\). Therefore, \(x = 75\).

a) \(x = 25\) b) \(x = 48\) c) \(x = 75\)

- Can a percent describe more than one whole? - How do you convert any percent to a decimal? - First write the percent as a fraction with denominator \(100\), then simplify.

1. Maya's statement is incorrect. Percents can be greater than \(100\%\), which represent values greater than \(1\). Since \(125\% = 1.25\), this value is greater than \(1\). 2. Ben's statement is correct. Divide by \(100\) to convert a percent to a decimal: \(125 \div 100 = 1.25\). 3. Carlos's statement is correct. Write \(125\%\) as \(\frac{125}{100}\), then divide the numerator and denominator by \(25\): \(\frac{125}{100} = \frac{5}{4}\).

Maya is incorrect: \(125\% = 1.25\), which is greater than \(1\). Ben is correct: \(125\% = 1.25\). Carlos is correct: \(125\% = \frac{125}{100} = \frac{5}{4}\).

- Compare both values in the same form. - Simplify a fraction first if that makes a denominator of one hundred easier. - Convert by dividing when you cannot conveniently write hundredths directly.

1. For a), \(\frac{3}{8}=37.5\%\), so \(\frac{3}{8}<40\%\). 2. For b), \(\frac{12}{60}=\frac{1}{5}=20\%\), so the values are equal. 3. For c), \(\frac{9}{12}=\frac{3}{4}=75\%\), so \(\frac{9}{12}>70\%\). 4. For d), \(\frac{7}{20}=\frac{35}{100}=35\%\), so \(\frac{7}{20}>34\%\).

a) \(\frac{3}{8}=37.5\%<40\%\) b) \(\frac{12}{60}=20\%=20\%\) c) \(\frac{9}{12}=75\%>70\%\) d) \(\frac{7}{20}=35\%>34\%\)

- Use the grandfather’s gift to find the full price. - Express one half as a fraction or percent. - Add all three contributions, then compare the amount left after the tablet purchase with \(\$14.50\).

1. The \(\$100\) gift is \(25\%\) of the tablet’s price. Scaling from \(25\%\) to \(100\%\), the price is \(\$100 \times 4 = \$400\). 2. Maya has saved \(30\%\) of \(\$400\): \(\$400 \times 0.30 = \$120\). 3. Her parents contribute one half of \(\$400\), which is \(\$200\). 4. The total available is \(\$120 + \$200 + \$100 = \$420\). 5. After buying the tablet, \(\$420 - \$400 = \$20\) remains. Because \(\$20 > \$14.50\), the remaining money is enough for the screen protector.

a) \(\$400\) b) Yes. Maya has \(\$420\) in all, so \(\$20\) remains after buying the tablet. That is more than the \(\$14.50\) cost of the screen protector.

- Convert one fourth to a percent. - Find the percent of the price represented by the remaining \(\$140\). - Scale from that part to \(100\%\).

1. One fourth is \(25\%\). 2. The first two amounts represent \(25\% + 35\% = 60\%\) of the price. 3. The remaining \(\$140\) represents \(100\% - 60\% = 40\%\) of the price. 4. Find the whole: \(\$140 \div 0.40 = \$350\).

The smartphone costs \(\$350\).

- Convert one fifth to a percent. - Find the percent left after lodging and meals. - Set the remaining percent equal to \(\$630\), then find \(100\%\).

1. One fifth is \(20\%\). 2. Lodging and meals use \(45\% + 20\% = 65\%\) of the budget. 3. Transportation and admission fees use \(100\% - 65\% = 35\%\). 4. Since \(35\%\) is \(\$630\), the whole budget is \(\$630 \div 0.35 = \$1800\).

The total cost is \(\$1800\).

- Use the known area and percent to find the whole garden area. - For a rectangle, how are area, length, and width related?

1. The flower bed is \(14\%\) of the garden, so the garden’s area is \(12.6\,\text{m}^2 \div 0.14 = 90\,\text{m}^2\). 2. For a rectangle, length equals area divided by width: \(90\,\text{m}^2 \div 15\,\text{m} = 6\,\text{m}\).

a) \(90\,\text{m}^2\) b) \(6\,\text{m}\)

- Find what percent of the capacity the added \(130\,\text{L}\) represents. - Use that part to find the whole capacity. - Then find the percent still missing from a full barrel.

1. The added water changed the level by \(60\% - 35\% = 25\%\) of the barrel’s capacity. 2. Since \(25\%\) is \(130\,\text{L}\), the full capacity is \(130\,\text{L} \div 0.25 = 520\,\text{L}\). 3. The barrel is missing \(100\% - 60\% = 40\%\) of its capacity. 4. The missing volume is \(520\,\text{L} \times 0.40 = 208\,\text{L}\).

a) \(520\,\text{L}\) b) \(208\,\text{L}\)

- What percent do all responses total? - What percent remains after subtracting the other two sports? - Write the remaining percent as a fraction, then simplify.

1. Add the percents for soccer and swimming: \(25\% + 40\% = 65\%\). 2. Subtract from \(100\%\) to find the percent who chose track and field: \(100\% - 65\% = 35\%\). 3. Write \(35\%\) as a fraction: \(35\% = \frac{35}{100}\). 4. Simplify by dividing the numerator and denominator by \(5\): \(\frac{35}{100} = \frac{7}{20}\).

\(\frac{7}{20}\)

- First write every value as a percent. - For each fraction, try to make the denominator \(100\). - Which two percent values are equal?

1. For A, \(\frac{7}{20} = \frac{35}{100} = 35\%\). 2. Value B is already written as \(34\%\). 3. For C, \(0.36 = 36\%\). 4. For D, \(\frac{18}{50} = \frac{36}{100} = 36\%\). 5. Values C and D are equal because both represent \(36\%\).

A: \(35\%\); B: \(34\%\); C: \(36\%\); D: \(36\%\). C and D represent the same portion.

- Write all the portions in the same form before adding. - How much is missing from \(100\%\)? - Once you have the decimal, how can you write it as a fraction and a percent?

1. Write the given values as decimals: \(\frac{1}{5} = 0.2\), \(0.35 = 0.35\), and \(15\% = 0.15\). 2. Add the known values: \(0.2 + 0.35 + 0.15 = 0.7\). 3. Subtract from \(1\): \(x = 1 - 0.7 = 0.3\). 4. Convert \(0.3\) to the other forms: \(0.3 = \frac{3}{10} = 30\%\).

Fraction: \(\frac{3}{10}\); decimal: \(0.3\); percent: \(30\%\)

- Apply the stated divisibility rule to the numerator and denominator. - Decide whether the resulting division gives a whole number or leaves a remainder.

1. Multiply the numerator by \(100\): \(5 \times 100 = 500\). 2. Check whether \(500\) is divisible by the denominator, \(14\). 3. Divide: \(500 \div 14 = 35\) remainder \(10\), or \(35.714\ldots\). 4. Because the division has a remainder, \(\frac{5}{14}\) cannot be written as a whole-number percent.

\(\frac{5}{14}\) cannot be written as a whole-number percent because \(500\) is not divisible by \(14\): \(500 \div 14 = 35\) remainder \(10\).

- What percent do all the materials total? - Convert \(\frac{3}{8}\) to a decimal or percent. - What percent is one tenth? - First find the total percent represented by sand and clay.

1. Convert the sand portion: \(\frac{3}{8} = 0.375 = 37.5\%\). 2. Convert the clay portion: \(0.45 = 45\%\). 3. Add the known portions: \(37.5\% + 45\% = 82.5\%\). 4. Subtract from the whole: \(100\% - 82.5\% = 17.5\%\). 5. Since \(\frac{1}{10} = 10\%\) and \(17.5\% > 10\%\), the humus portion is greater than \(\frac{1}{10}\).

The humus portion is \(17.5\%\). Yes, it is greater than \(\frac{1}{10}\), which is \(10\%\).

- A percent is a count for every hundred. - Think about fractions equivalent to hundredths. - A percent can exceed one hundred, just as a decimal can exceed one.

1. \(\frac{4}{25}=\frac{16}{100}=0.16=16\%\). 2. \(0.08=8\%=\frac{8}{100}=\frac{2}{25}\). 3. \(\frac{13}{20}=\frac{65}{100}=0.65=65\%\). 4. \(120\%=1.2=\frac{120}{100}=\frac{6}{5}\).

<table><tr><th>Fraction</th><th>Decimal</th><th>Percent</th></tr><tr><td>\(\frac{4}{25}\)</td><td>\(0.16\)</td><td>\(16\%\)</td></tr><tr><td>\(\frac{2}{25}\)</td><td>\(0.08\)</td><td>\(8\%\)</td></tr><tr><td>\(\frac{13}{20}\)</td><td>\(0.65\)</td><td>\(65\%\)</td></tr><tr><td>\(\frac{6}{5}\)</td><td>\(1.2\)</td><td>\(120\%\)</td></tr></table>

- How do you write \(0.28\) directly as a percent? - Write all the portions in the same form before comparing or adding. - Add your results and compare the sum with \(100\%\).

1. Bicycle: \(0.28 = 28\%\). 2. Walking: \(\frac{7}{25} = \frac{28}{100} = 28\%\). 3. Bus: \(\frac{9}{40} = 0.225 = 22.5\%\). 4. Car: \(\frac{105}{500} = \frac{21}{100} = 21\%\). 5. Add the percents: \(28\% + 28\% + 22.5\% + 21\% = 99.5\%\). 6. Subtract from the whole: \(100\% - 99.5\% = 0.5\%\).

a) Bicycle: \(28\%\); walking: \(28\%\); bus: \(22.5\%\); car: \(21\%\) b) The total is \(99.5\%\), so \(0.5\%\) is missing.

- Percent means “per hundred.” - Convert a fraction to a decimal by dividing the numerator by the denominator. - For phrases such as “seven out of ten,” first write a fraction.

1. For part a), \(\frac{1}{8} = 0.125 = 12.5\%\). 2. For part b), \(\frac{3}{20} = 0.15 = 15\%\). 3. For part c), \(\frac{7}{10} = 0.7 = 70\%\). 4. For part d), \(2 \times 1.5\% = 3\% = 0.03\). 5. For part e), \(\frac{1}{4} \times 10\% = 2.5\% = 0.025\).

a) \(0.125\) and \(12.5\%\) b) \(0.15\) and \(15\%\) c) \(0.7\) and \(70\%\) d) \(0.03\) and \(3\%\) e) \(0.025\) and \(2.5\%\)

- Find the total collection before finding the fantasy category. - Use the known part and percent to find \(100\%\). - Then find \(20\%\) of the total.

1. Since \(45\) is \(15\%\) of the collection, the total is \(45 \div 0.15 = 300\) books. 2. Fantasy books make up \(20\%\) of \(300\): \(300 \times 0.20 = 60\).

There are \(300\) books in the collection and \(60\) fantasy books.

- Find each whole separately. - Consider how the whole changes when the same part represents a smaller percent.

1. Divide \(24\) by each percent written as a decimal. 2. a) \(24 \div 0.50 = 48\) 3. b) \(24 \div 0.20 = 120\) 4. c) \(24 \div 0.03 = 800\) 5. Therefore, \(48 < 120 < 800\).

a) \(G = 48\) b) \(G = 120\) c) \(G = 800\) Order: \(48 < 120 < 800\)

- Keep each unit in the answer. - It may help to write \(75\%\) as \(\frac{3}{4}\). - What simple fraction is equivalent to \(12.5\%\)?

1. Divide each part by the percent written as a decimal. 2. a) \(x = \$45 \div 0.15 = \$300\) 3. b) \(x = 90\,\text{kg} \div 0.75 = 120\,\text{kg}\) 4. c) \(x = 10 \div 0.125 = 80\) 5. d) \(x = 14\,\text{m} \div 0.40 = 35\,\text{m}\)

a) \(x = \$300\) b) \(x = 120\,\text{kg}\) c) \(x = 80\) d) \(x = 35\,\text{m}\)

- Convert both “one out of every” rates to percents. - Does the current rate show an increase or decrease from ten years ago? - How are percentage points different from the number of students? - Compare today's rate with the goal, rather than with the rate ten years ago.

1. The old rate was \(\frac{1}{10}=10\%\). 2. The current rate is \(\frac{1}{8}=12.5\%\), higher than before. 3. The rate rose by \(12.5-10=2.5\) percentage points. 4. The goal is \(20\%-12.5\%=7.5\) percentage points above today's rate.

a) Ten years ago: \(10\%\); today: \(12.5\%\) b) \(2.5\) percentage points c) \(7.5\) percentage points

- Find \(100\%\) of each person’s savings goal separately. - Compare the two prices, then subtract to find the difference.

1. Noah’s goal is \(\$108 \div 0.45 = \$240\). 2. Ava’s goal is \(\$140 \div 0.70 = \$200\). 3. Since \(\$240 > \$200\), Noah plans to buy the more expensive bike. 4. The price difference is \(\$240 - \$200 = \$40\).

Noah’s bike costs \(\$240\), Ava’s costs \(\$200\), and the difference is \(\$40\). Noah plans to buy the more expensive bike.

- Simplify \(\frac{100}{40}\) first. - When is \(2.5x\) a whole number? - Test several small values of \(x\) and look for a pattern.

1. Convert the fraction to a percent: \(\frac{x}{40} \times 100 = \frac{100x}{40}\). 2. Simplify: \(\frac{100x}{40} = \frac{5x}{2} = 2.5x\). 3. For \(\frac{5x}{2}\) to be a whole number, \(x\) must be divisible by \(2\), so \(x\) must be even. 4. For example, if \(x = 2\), then \(\frac{2}{40} = 5\%\). If \(x = 4\), then \(\frac{4}{40} = 10\%\). 5. If \(x = 1\), then \(\frac{1}{40} = 2.5\%\). If \(x = 3\), then \(\frac{3}{40} = 7.5\%\).

a) \(\frac{x}{40} \times 100 = \frac{5x}{2}\). This is a whole number only when \(x\) is even. b) Examples that give whole-number percents: \(x = 2\), which gives \(5\%\), and \(x = 4\), which gives \(10\%\). Examples that do not: \(x = 1\), which gives \(2.5\%\), and \(x = 3\), which gives \(7.5\%\).

- Identify which quantity is missing in each row. - Think about how the part and the whole relate to the percent. - For the last row, find the whole from a known small share.

1. The missing part is \(\$450\times0.12=\$54\). 2. The missing percent is \(\frac{20}{80}\times100\%=25\%\). 3. Since \(5\%\) of the missing whole is \(15\,\text{m}\), the whole is \(15\div0.05=300\,\text{m}\).

<table><tr><th>Whole</th><th>Percent</th><th>Part</th></tr><tr><td>\(\$450\)</td><td>\(12\%\)</td><td>\(\$54\)</td></tr><tr><td>\(80\,\text{kg}\)</td><td>\(25\%\)</td><td>\(20\,\text{kg}\)</td></tr><tr><td>\(300\,\text{m}\)</td><td>\(5\%\)</td><td>\(15\,\text{m}\)</td></tr></table>

- First find the total number of club members. - Then subtract the soccer group, or find the remaining percent.

1. Find the total membership: \(54 \div 0.45 = 120\). 2. Subtract the soccer members: \(120 - 54 = 66\). 3. Equivalently, the other sports represent \(100\% - 45\% = 55\%\), and \(120 \times 0.55 = 66\).

\(66\) members participate in other sports.

- Find the total number of votes from Jordan's share. - Think about how votes relate to voters in this election. - All three candidates' votes account for the whole election.

1. The total number of votes is \(9\div0.30=30\). Since each student casts one vote, \(30\) students voted. 2. Casey's share is \(6\div30=0.20=20\%\). 3. Riley receives \(30-9-6=15\) votes, representing \(15\div30=0.50=50\%\).

a) \(30\) students b) \(20\%\) c) \(15\) votes; \(50\%\)

- Use the nonfiction category to find the entire collection. - Once you know the total, find \(35\%\) of it. - For the final category, find the percent left after the first two categories.

1. Since \(108\) is \(12\%\) of the collection, the total is \(108 \div 0.12 = 900\) books. 2. The number of novels is \(900 \times 0.35 = 315\). 3. Graphic novels represent \(100\% - 12\% - 35\% = 53\%\). 4. Their number is \(900 - 108 - 315 = 477\).

a) \(900\) books b) \(315\) novels c) \(477\) graphic novels, or \(53\%\) of the collection

- What percent do all four groups total? - How can you write a decimal directly as hundredths? - Write all the given portions with denominator \(100\).

1. Convert the given portions: instrument, \(\frac{1}{5} = \frac{20}{100} = 20\%\); sports team, \(0.32 = 32\%\); reading, \(\frac{6}{25} = \frac{24}{100} = 24\%\). 2. Add the known percents: \(20\% + 32\% + 24\% = 76\%\). 3. Subtract from the whole: \(100\% - 76\% = 24\%\).

Instrument: \(20\%\) Sports team: \(32\%\) Reading: \(24\%\) Other: \(24\%\)

- Put each fraction and percent in a common form before comparing them. - For a repeating fraction, keep enough digits to decide the comparison without treating the decimal as exact. - In part c), distinguish the portion eaten from the portion that remains.

1. For a), \(\frac{1}{3} = 0.333\ldots = 33.333\ldots\%\). Since \(33.333\ldots\% > 33\%\), the statement is true. 2. For b), \(\frac{5}{8} = 0.625 = 62.5\%\). The statement is true. 3. For c), \(\frac{1}{12} = 0.08333\ldots = 8.333\ldots\%\). The remaining portion is \(100\% - 8.333\ldots\% = 91.666\ldots\%\). Since this is greater than \(90\%\), the statement is false.

a) True, because \(\frac{1}{3} = 33.333\ldots\% > 33\%\). b) True, because \(\frac{5}{8} = 0.625 = 62.5\%\). c) False. \(91.666\ldots\%\) remains, which is greater than \(90\%\).

- Think about how a fraction can be represented as a decimal before writing it as a percent. - Keep enough decimal places to determine the nearest tenth of a percent. - Check that each rounded percent is reasonable for the size of the original fraction.

1. \(5 \div 12 = 0.41666\ldots\), so \(\frac{5}{12} = 41.666\ldots\% \approx 41.7\%\). 2. \(4 \div 7 = 0.571428\ldots\), so \(\frac{4}{7} = 57.1428\ldots\% \approx 57.1\%\). 3. \(11 \div 15 = 0.73333\ldots\), so \(\frac{11}{15} = 73.333\ldots\% \approx 73.3\%\).

1. \(\approx 41.7\%\) 2. \(\approx 57.1\%\) 3. \(\approx 73.3\%\)

- Write each portion as a fraction first. - Divide the numerator by the denominator to find the decimal. - Multiply the decimal by \(100\) to find the percent.

1. For part a, \(\frac{12}{40} = \frac{3}{10} = 0.3 = 30\%\). 2. For part b, \(\frac{1}{8} = 0.125 = 12.5\%\). 3. For part c, \(\frac{15}{2000} = 0.0075 = 0.75\%\). 4. For part d, \(\frac{3}{8} = 0.375 = 37.5\%\).

a) \(0.3\), \(30\%\) b) \(0.125\), \(12.5\%\) c) \(0.0075\), \(0.75\%\) d) \(0.375\), \(37.5\%\)

- For part a, find the total mass first. - For part c, convert the measurements to the same unit. - A scale of \(1 : 200\) means a model length is \(\frac{1}{200}\) of the corresponding full-size length.

1. For part a, the total mass is \(2\,\text{g} + 498\,\text{g} = 500\,\text{g}\). The gold portion is \(\frac{2}{500} = 0.004 = 0.4\%\). 2. For part b, the scale portion is \(\frac{1}{200} = 0.005 = 0.5\%\). 3. For part c, \(1\,\text{L} = 1000\,\text{mL}\). The syrup portion is \(\frac{5}{1000} = 0.005 = 0.5\%\).

a) \(0.004\), \(0.4\%\) b) \(0.005\), \(0.5\%\) c) \(0.005\), \(0.5\%\)

- If \(20\%\) is \(48\), how can you use that information to find \(100\%\)? - Can you find the answer to part b) without first finding the whole? - Think about how the part changes when the percent changes but the whole stays the same.

1. The whole number is \(48 \div 0.20 = 240\). 2. Then \(75\%\) of the whole is \(240 \times 0.75 = 180\). 3. To check Nimal's claim, compute \(48 \times 3.75 = 180\). This matches the result from part b), so the claim is correct.

a) \(240\) b) \(180\) c) Nimal's claim is correct because \(48 \times 3.75 = 180\).

- Identify the full capacity and the amount of charge in each part. - How do you find a percent when the part and the whole are known? - For c), first find the change in the battery percentage. - Include the correct unit in each charge amount.

1. The current percent is \(\frac{1125}{4500} = 0.25 = 25\%\). 2. At \(20\%\), the remaining charge is \(4500\,\text{mAh} \times 0.20 = 900\,\text{mAh}\). 3. The battery level drops by \(20 - 5 = 15\) percentage points. This amount is \(15\%\) of the full capacity, so the charge used is \(4500\,\text{mAh} \times 0.15 = 675\,\text{mAh}\).

a) \(25\%\) b) \(900\,\text{mAh}\) c) \(675\,\text{mAh}\)

- When a part and its percent are known, what calculation finds the whole? - The percentages for all tree types must add to \(100\%\). - After finding the total, can you subtract the number of oaks to find the number of beeches?

1. The total number of trees is \(420 \div 0.35 = 1200\). 2. The percent that are beeches is \(100\% - 35\% = 65\%\). 3. The number of beeches is \(1200 - 420 = 780\). Equivalently, \(1200 \times 0.65 = 780\).

a) \(1200\) trees b) \(65\%\) c) \(780\) beech trees

- Use the number of instrument players and its percent to find the total number surveyed. - After finding the total, multiply it by the percent who play on a sports team. - For c), update the number of instrument players but keep the same total.

1. The total number surveyed is \(162 \div 0.27 = 600\). 2. The number who play on a sports team is \(600 \times 0.45 = 270\). 3. The new number who play an instrument is \(162 + 18 = 180\). 4. The new percent is \(\frac{180}{600} = 0.30 = 30\%\).

a) \(600\) students b) \(270\) students c) \(30\%\)

- When the part and percent are known, divide the part by the percent written as a decimal. - Find the two wholes separately before comparing them.

1. For Case 1, the whole is \(35 \div 0.14 = 250\). 2. For Case 2, the whole is \(60 \div 0.25 = 240\). 3. Since \(250 > 240\), the whole in Case 1 is greater.

Case 1: \(250\) Case 2: \(240\) The whole in Case 1 is greater.

- Read the graphic-novel percent from the graph. - Use the known \(15\%\) part to find \(100\%\). - Then find \(45\%\) of the total.

1. Graphic novels represent \(15\%\) of the collection. 2. Since \(90\) is \(15\%\), the total number of books is \(90 \div 0.15 = 600\). 3. Nonfiction books represent \(45\%\) of the collection: \(600 \times 0.45 = 270\).

a) \(600\) books b) \(270\) nonfiction books

- Find groceries' share on the chart before using its dollar amount. - Which amount represents the full budget? - Use the graph's other percentages with that total.

1. The graph shows groceries take \(30\%\) of the budget, so the total is \(\$450\div0.30=\$1500\). 2. Rent is \(40\%\) of \(\$1500\), or \(\$600\). 3. Entertainment and other expenses are each \(15\%\), or \(\$225\) each.

a) \(\$1500\) b) Rent: \(\$600\); entertainment: \(\$225\); other: \(\$225\)

- Find the sports percentage on the graph. - Use that percentage with the known number of students to find the whole. - Use the other sectors to find their student counts.

1. The graph shows that sports is \(30\%\) of the survey, so the total is \(120\div0.30=400\) students. 2. Gaming is \(40\%\) of \(400\), or \(160\) students. 3. Music is \(20\%\), or \(80\), and reading is \(10\%\), or \(40\).

a) \(400\) students b) Gaming: \(160\); music: \(80\); reading: \(40\)

- Use the graph to see what fraction of the entire group cycles. - Is the entire student body represented by this survey? - Find the whole group before finding the other travel groups.

1. The graph shows bicycle riders make up \(25\%\). The school has \(120\div0.25=480\) students. 2. Bus: \(40\%\) of \(480\) is \(192\). 3. Walking: \(20\%\) of \(480\) is \(96\). 4. Car: \(15\%\) of \(480\) is \(72\).

a) \(480\) students b) Bus: \(192\); walking: \(96\); car: \(72\)

- Count the total tiles and the shaded border tiles. - Express the shaded portion as a fraction or percent of the grid. - Use the shaded area to find the whole area.

1. The grid has \(5 \times 5 = 25\) equal tiles, and \(16\) border tiles are shaded. 2. The installed border is \(\frac{16}{25} = 0.64 = 64\%\) of the patio. 3. Since \(4.48\,\text{m}^2\) is \(64\%\) of the total, the patio area is \(4.48\,\text{m}^2 \div 0.64 = 7\,\text{m}^2\).

The entire patio has an area of \(7\,\text{m}^2\).

- You know the number represented by \(15\%\). Use it to find the whole, or \(100\%\). - After finding the total number of songs, multiply it by the pop percentage. - A percent table can help: if \(15\%\) corresponds to \(12\), what corresponds to \(1\%\)?

1. Jazz songs make up \(15\%\) of the total. Therefore, the total number of songs is \(12 \div 0.15 = 80\). 2. Pop songs make up \(40\%\) of the total, so the number of pop songs is \(80 \times 0.40 = 32\).

a) \(80\) songs b) \(32\) pop songs

- Count the equal parts in each whole to find the denominator. - Count the shaded parts to find the numerator. - Write each fraction with denominator \(10\), \(100\), or \(1000\) when possible. - Percent means “per hundred.”

1. Count the shaded parts and total parts: a) \(\frac{11}{20}\) b) \(\frac{7}{10}\) c) \(\frac{8}{25}\) 2. Convert to decimals: a) \(\frac{11}{20} = \frac{55}{100} = 0.55\) b) \(\frac{7}{10} = 0.7\) c) \(\frac{8}{25} = \frac{32}{100} = 0.32\) 3. Convert to percents: a) \(0.55 = 55\%\) b) \(0.7 = 70\%\) c) \(0.32 = 32\%\)

a) \(\frac{11}{20} = 0.55 = 55\%\) b) \(\frac{7}{10} = 0.7 = 70\%\) c) \(\frac{8}{25} = 0.32 = 32\%\)

- Write the percent and the fraction with compatible forms. - Find the fraction of the budget left for sports equipment. - Use the value of that fraction to find the whole budget.

1. Write \(37.5\%\) as a fraction: \(37.5\% = \frac{3}{8}\). 2. Paint, wall coverings, and furniture use \(\frac{3}{8} + \frac{1}{3} = \frac{9}{24} + \frac{8}{24} = \frac{17}{24}\) of the budget. 3. Sports equipment uses \(1 - \frac{17}{24} = \frac{7}{24}\) of the budget. 4. Since \(\frac{7}{24}\) equals \(\$1050\), the whole budget is \(\$1050 \times \frac{24}{7} = \$3600\).

The total renovation budget was \(\$3600\).

- Multiply the numerator by \(100\) and check whether the result is divisible by \(16\). - Can you simplify \(\frac{100}{16}\) first? - Look for a pattern in the values of \(k\) that work.

1. For the percent to be a whole number, \(\frac{100k}{16}\) must be a whole number. 2. Simplify the expression: \(\frac{100k}{16} = \frac{25k}{4}\). 3. Since \(25\) is not divisible by \(4\), \(k\) must be a multiple of \(4\). 4. The multiples of \(4\) from \(1\) through \(10\) are \(4\) and \(8\). 5. Check: \(\frac{4}{16} = 25\%\) and \(\frac{8}{16} = 50\%\).

The values are \(k = 4\) and \(k = 8\). For these values, \(100k\) is divisible by \(16\): \(400 \div 16 = 25\) and \(800 \div 16 = 50\).

- First use \(\$18\) and \(60\%\) to find the original savings. - Add \(\$10\), then find what percent \(\$18\) is of the new amount. - Compare that percent with \(45\%\).

1. Since \(\$18\) is \(60\%\) of the savings, the actual savings were \(\$18 \div 0.60 = \$30\). 2. With \(\$10\) more, the savings would have been \(\$30 + \$10 = \$40\). 3. The game would then represent \(\frac{18}{40} \times 100\% = 45\%\). 4. This matches the claim, so the statement is correct.

Alex had saved \(\$30\). The statement is correct because \(\$18\) is \(45\%\) of \(\$40\).

- Write “one out of every eight,” “half,” and “one out of every twenty” as percents. - Compare those values with the percents stated in the report. - Check whether the phrase “nearly half” matches the actual value.

1. “One out of every eight” means \(\frac{1}{8} = 12.5\%\), not \(25\%\). 2. Half is \(50\%\). A value of \(12.5\%\) is not nearly half. 3. A goal of \(10\%\) means \(\frac{10}{100} = \frac{1}{10}\), or one out of every ten households. One out of every twenty is \(\frac{1}{20} = 5\%\). 4. A corrected report is: “In our city, one out of every eight households, or \(12.5\%\), does not sort its recycling correctly. We need to reach our goal of \(10\%\), or one out of every ten households.”

The errors are: 1. One out of every eight is \(12.5\%\), not \(25\%\). 2. \(12.5\%\) is not nearly half, which is \(50\%\). 3. \(10\%\) means one out of every ten, not one out of every twenty, which is \(5\%\). Corrected report: “In our city, one out of every eight households, or \(12.5\%\), does not sort its recycling correctly. We need to reach our goal of \(10\%\), or one out of every ten households.”

- Simplify both fractions completely first. - Think about which denominators can be scaled to \(100\) using a whole-number factor. - Compare the simplified denominators with the factors of \(100\).

1. Simplify \(\frac{6}{15}\) by dividing by \(3\): \(\frac{6}{15} = \frac{2}{5}\). Since \(5\) is a factor of \(100\), multiply by \(\frac{20}{20}\): \(\frac{2}{5} = \frac{40}{100} = 40\%\). 2. Simplify \(\frac{6}{14}\) by dividing by \(2\): \(\frac{6}{14} = \frac{3}{7}\). Since \(7\) is not a factor of \(100\), the fraction cannot be scaled to a fraction with denominator \(100\) using whole numbers. Its percent is \(42.857\ldots\%\), not a whole number. 3. The difference is whether the denominator of the fraction in simplest form is a factor of \(100\).

\(\frac{6}{15} = \frac{2}{5} = 40\%\) because \(5\) is a factor of \(100\). \(\frac{6}{14} = \frac{3}{7} = 42.857\ldots\%\), which is not a whole-number percent because \(7\) is not a factor of \(100\). The denominator of the fraction in simplest form determines whether it can be written as an exact whole-number percent.