Free Math Worksheets

- 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.

- 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. \(0.004 = \frac{4}{1000} = \frac{1}{250} = 0.4\%\).

Row 1: \(0.4\); \(40\%\) Row 2: \(\frac{2}{25}\); \(8\%\) Row 3: \(\frac{3}{20}\); \(0.15\) Row 4: \(\frac{1}{250}\); \(0.4\%\)

- For a fraction, divide the numerator by the denominator. - How many places does the decimal point move when you convert to a percent? - Use the hundredths digit 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.

- Try to write each fraction with denominator \(100\). - What must be true of the denominator for the fraction to become an exact number of hundredths? - You can multiply the numerator by \(100\), then divide by the denominator.

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 \(30\)

- First decide whether each missing value is the whole, the percent, or the part. - Use the relationship: part \(=\) percent written as a decimal \(\times\) whole. - You can write the percent as a decimal or as a fraction to make the calculation easier.

1. The missing part is \(\$450 \times 0.12 = \$54\). 2. The missing percent is \(\frac{20\,\text{kg}}{80\,\text{kg}} = 0.25 = 25\%\). 3. The missing whole is \(15\,\text{m} \div 0.05 = 300\,\text{m}\).

1. \(\$54\) 2. \(25\%\) 3. \(300\,\text{m}\)

- 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 total number of small squares. - Count the shaded squares. - A percent describes a portion out of \(100\). - First write the portion as a fraction with denominator \(100\). - Simplify the fraction completely.

1. The grid has \(10\) rows and \(10\) columns, so it contains \(10 \times 10 = 100\) small squares. 2. Count the shaded squares: \(28\) squares are orange. 3. Since \(28\) out of \(100\) squares are shaded, the shaded portion is \(28\%\). 4. As a fraction, the shaded portion is \(\frac{28}{100}\). Divide the numerator and denominator by \(4\): \(\frac{28}{100} = \frac{7}{25}\).

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) \(30\%\)

- 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, \(0.12 \times 100 = 12\%\).

First row: \(\frac{7}{20}\) and \(0.35\) Second row: \(80\%\) and \(0.8\) Third row: \(12\%\) and \(\frac{3}{25}\)

- 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: \(125\% = \frac{125}{100} = \frac{5}{4}\).

- Write both values in each pair in the same form. - Some fractions can be simplified before you write them with denominator \(100\). - When an equivalent fraction with denominator \(100\) is not convenient, divide the numerator by the denominator.

1. For a), \(3 \div 8 = 0.375 = 37.5\%\). Since \(37.5\% < 40\%\), \(\frac{3}{8} < 40\%\). 2. For b), simplify \(\frac{12}{60}\) to \(\frac{1}{5}\), then write \(\frac{1}{5} = \frac{20}{100} = 20\%\). Therefore, \(\frac{12}{60} = 20\%\). 3. For c), simplify \(\frac{9}{12}\) to \(\frac{3}{4}\), then write \(\frac{3}{4} = \frac{75}{100} = 75\%\). Since \(75\% > 70\%\), \(\frac{9}{12} > 70\%\). 4. For d), \(1 \div 3 = 0.333\ldots\), so \(\frac{1}{3} = 33.333\ldots\%\). Since \(33.333\ldots\% > 33\%\), \(\frac{1}{3} > 33\%\).

a) \(<\) b) \(=\) c) \(>\) d) \(>\)

- 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: \(30\%\)

- What does it mean for one number to be divisible by another? - Divide \(500\) by \(14\) and check whether there is 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 denominator of \(100\) is useful when converting between fractions and percents. - How are hundredths related to two decimal places?

1. Row 1: \(\frac{4}{25} = \frac{16}{100}\), so the decimal is \(0.16\) and the percent is \(16\%\). 2. Row 2: \(0.08 = 8\%\). As a fraction, \(0.08 = \frac{8}{100} = \frac{2}{25}\). 3. Row 3: \(\frac{13}{20} = \frac{65}{100}\), so the decimal is \(0.65\) and the percent is \(65\%\). 4. Row 4: \(120\% = 1.20 = 1.2\). As a fraction, \(120\% = \frac{120}{100} = \frac{6}{5}\).

Row 1: \(0.16\); \(16\%\) Row 2: \(\frac{2}{25}\); \(8\%\) Row 3: \(0.65\); \(65\%\) Row 4: \(\frac{6}{5}\); \(1.2\)

- 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) \(x = 35\,\text{m}\)

- Find the percents for one out of every ten and one out of every eight. - Does a larger portion represent an improvement in this situation? - Is a goal of \(5\%\) higher or lower than the current value?

1. One out of every ten is \(\frac{1}{10} = 10\%\). One out of every eight is \(\frac{1}{8} = 12.5\%\). 2. An increase from \(10\%\) to \(12.5\%\) is an improvement, but the word “only” suggests a decrease. 3. A goal of \(5\%\) is lower than both \(10\%\) and \(12.5\%\), so it would not represent improvement. The word “again” also conflicts with the earlier value of \(10\%\). 4. One possible revision is: “Ten years ago, one out of every ten children, or \(10\%\), read a book every day. Today the situation has improved because one out of every eight children, or \(12.5\%\), reads a book every day. Our goal is for one out of every five children, or \(20\%\), to read a book every day.”

The statement is illogical because one out of every eight, or \(12.5\%\), is greater than one out of every ten, or \(10\%\), so “only” does not fit. Also, a goal of \(5\%\) would be lower than both earlier values. Revised statement: “Ten years ago, one out of every ten children, or \(10\%\), read a book every day. Today the situation has improved because one out of every eight children, or \(12.5\%\), reads a book every day. Our goal is for one out of every five children, or \(20\%\), to read a book every day.”

- 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 \(7.5\%\).

- 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.

- Use the known \(30\%\) share to find all votes. - The vote counts must total the number found in part a). - The percentages must total \(100\%\).

1. Since \(9\) votes is \(30\%\) of the total, the total number of votes is \(9 \div 0.30 = 30\). 2. Casey’s share is \(\frac{6}{30} \times 100\% = 20\%\). 3. Riley received \(30 - 9 - 6 = 15\) votes. 4. Riley’s share is \(\frac{15}{30} \times 100\% = 50\%\).

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

- 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: \(24\%\)

- Divide the numerator by the denominator when the fraction cannot easily be written with denominator \(100\). - What happens when you divide \(1\) by \(3\)? - Subtract the percent eaten from \(100\%\) to find the percent remaining.

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 \(90\%\).

- Divide the numerator by the denominator when the fraction cannot easily be written with denominator \(100\). - Use the usual rounding rule for the hundredths digit of the percent. - Calculate enough decimal places to round to the nearest tenth of a percent.

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 the claim, compute \(48 \times 3.75 = 180\). This matches the result from part b), so the claim is correct.

a) \(240\) b) \(180\) c) The 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

- Use the grocery amount and its percent to find the whole budget. - Once you know the total, multiply it by each remaining percent.

1. Since \(\$450\) is \(30\%\) of the budget, the total is \(\$450 \div 0.30 = \$1500\). 2. Rent: \(\$1500 \times 0.40 = \$600\). 3. Entertainment: \(\$1500 \times 0.15 = \$225\). 4. Other expenses: \(\$1500 \times 0.15 = \$225\).

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

- Match the \(120\) students with the sports percent. - Use that part to find the whole survey group. - Then multiply the total by each other percent.

1. Since \(120\) is \(30\%\) of the group, the total is \(120 \div 0.30 = 400\) students. 2. Gaming: \(400 \times 0.40 = 160\). 3. Music: \(400 \times 0.20 = 80\). 4. Reading: \(400 \times 0.10 = 40\).

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

- Read the bicycle percent from the graph. - Use the \(120\) bicycle riders to find the total number of students. - Check that all four travel groups add to the total.

1. Bicycle riders represent \(25\%\) of the students, so the total enrollment is \(120 \div 0.25 = 480\). 2. Bus: \(480 \times 0.40 = 192\). 3. Walking: \(480 \times 0.20 = 96\). 4. Car: \(480 \times 0.15 = 72\).

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. - What are the prime factors of \(100\)? - Can a fraction with denominator \(7\) be scaled to denominator \(100\) using whole numbers?

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.

- 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