Aimathic – Generate equivalent fractions and simplify

Generate equivalent fractions and simplify

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Difficulty: 1 – 5

5100524

Which fraction is already in simplest form? a) $\frac{15}{21}$ b) $\frac{12}{25}$ c) $\frac{35}{42}$ d) $\frac{24}{45}$

Hints

- Check whether the numerator and denominator share a factor greater than $1$. - Use divisibility rules for $2$, $3$, $5$, and $7$. - A fraction is in simplest form when its numerator and denominator have no common factor greater than $1$.

Solution

1. In a), $15$ and $21$ are both divisible by $3$, so the fraction can be simplified. 2. In b), $12$ and $25$ have no common factor greater than $1$, so the fraction is in simplest form. 3. In c), $35$ and $42$ are both divisible by $7$. 4. In d), $24$ and $45$ are both divisible by $3$. 5. Therefore, only $\frac{12}{25}$ is in simplest form.

Answer

b) $\frac{12}{25}$

5101934

Find the missing values $x$, $y$, and $z$. a) $\frac{4}{9}=\frac{x}{54}$ b) $\frac{12}{17}=\frac{60}{y}$ c) $\frac{z}{25}=\frac{112}{175}$

Hints

- Determine the factor that changes the known numerator or denominator. - Apply the same factor to the other part of the fraction. - Equivalent fractions are formed by multiplying or dividing the numerator and denominator by the same nonzero number.

Solution

1. For a), $54\div9=6$, so multiply the numerator by $6$: $4\times6=24$. Thus, $x=24$. 2. For b), $60\div12=5$, so multiply the denominator by $5$: $17\times5=85$. Thus, $y=85$. 3. For c), $175\div25=7$. Since $z\times7=112$, $z=112\div7=16$.

Answer

a) $x=24$ b) $y=85$ c) $z=16$

5101944

Find the scale factor $k$ used to create each equivalent fraction. a) $\frac{5}{6}\rightarrow\frac{55}{66}$ b) $\frac{3}{14}\rightarrow\frac{39}{182}$ c) $\frac{8}{15}\rightarrow\frac{120}{225}$

Hints

- Divide the new numerator by the original numerator. - Check that the same factor changes the denominator. - Equivalent fractions use the same scale factor for the numerator and denominator.

Solution

1. For a), $55\div5=11$, and $6\times11=66$. Therefore, $k=11$. 2. For b), $39\div3=13$, and $14\times13=182$. Therefore, $k=13$. 3. For c), $120\div8=15$, and $15\times15=225$. Therefore, $k=15$.

Answer

a) $k=11$ b) $k=13$ c) $k=15$

5101964

Find the missing numerator in each equivalent fraction. a) $\frac{3}{5}=\frac{?}{20}$ b) $\frac{7}{8}=\frac{?}{56}$ c) $\frac{12}{13}=\frac{?}{39}$ d) $\frac{15}{4}=\frac{?}{100}$

Hints

- Divide the new denominator by the original denominator to find the scale factor. - Multiply the numerator by the same factor. - Check that the two fractions have the same value.

Solution

1. For a), $20\div5=4$, so $3\times4=12$. 2. For b), $56\div8=7$, so $7\times7=49$. 3. For c), $39\div13=3$, so $12\times3=36$. 4. For d), $100\div4=25$, so $15\times25=375$.

Answer

a) $12$ b) $49$ c) $36$ d) $375$

5101994

Divide the numerator and denominator of each fraction by the given number. Write “not possible” if the given number does not divide both the numerator and denominator. a) $\frac{60}{15}$ by $15$ b) $\frac{112}{14}$ by $14$ c) $\frac{144}{60}$ by $12$ d) $\frac{200}{75}$ by $25$ e) $\frac{130}{45}$ by $13$

Hints

- Reducing by a given number means dividing both the numerator and denominator by that number. - Both divisions must result in whole numbers. - Test the numerator and denominator separately.

Solution

1. For a), $60\div15=4$ and $15\div15=1$, so the result is $\frac{4}{1}=4$. 2. For b), $112\div14=8$ and $14\div14=1$, so the result is $8$. 3. For c), $144\div12=12$ and $60\div12=5$, so the result is $\frac{12}{5}$. 4. For d), $200\div25=8$ and $75\div25=3$, so the result is $\frac{8}{3}$. 5. For e), $130$ is divisible by $13$, but $45$ is not. The reduction is not possible.

Answer

a) $4$ b) $8$ c) $\frac{12}{5}$ d) $\frac{8}{3}$ e) not possible

5102024

Fill in each box so the fractions are equivalent. a) $\frac{2}{5}=\frac{14}{\Box}$ b) $\frac{\Box}{9}=\frac{54}{81}$ c) $\frac{72}{96}=\frac{3}{\Box}$ d) $\frac{13}{17}=\frac{\Box}{51}$

Hints

- Find the factor between the two known numerators or denominators. - Apply the same multiplication or division to the other part of the fraction. - Check that each pair represents the same value.

Solution

1. For a), $14\div2=7$, so the denominator is $5\times7=35$. 2. For b), $81\div9=9$, so the numerator is $54\div9=6$. 3. For c), $72\div3=24$, so the denominator is $96\div24=4$. 4. For d), $51\div17=3$, so the numerator is $13\times3=39$.

Answer

a) $35$ b) $6$ c) $4$ d) $39$

5102054

Write each part-to-whole relationship as a fraction in simplest form. a) $28$ of $70$ students in a grade play a musical instrument. b) $15$ of $45$ tickets in a prize drawing are winning tickets. c) $54$ of $72$ participants passed an assessment. d) $13$ of the $52$ cards in a standard deck belong to one suit.

Hints

- Write the number in the part over the total number. - Find a common factor of the numerator and denominator. - Continue simplifying until the numerator and denominator share no factor greater than $1$.

Solution

1. For a), $\frac{28}{70}=\frac{2}{5}$. 2. For b), $\frac{15}{45}=\frac{1}{3}$. 3. For c), $\frac{54}{72}=\frac{3}{4}$. 4. For d), $\frac{13}{52}=\frac{1}{4}$.

Answer

a) $\frac{2}{5}$ b) $\frac{1}{3}$ c) $\frac{3}{4}$ d) $\frac{1}{4}$

5102084

Write each fraction in simplest form. a) $\frac{45}{120}$ b) $\frac{56}{196}$ c) $\frac{144}{216}$

Hints

- Look for a common factor of the numerator and denominator. - You may simplify in one step or in several steps. - Stop when no common factor greater than $1$ remains.

Solution

1. For a), divide the numerator and denominator by $15$: $\frac{45}{120}=\frac{3}{8}$. 2. For b), divide by $28$: $\frac{56}{196}=\frac{2}{7}$. 3. For c), divide by $72$: $\frac{144}{216}=\frac{2}{3}$.

Answer

a) $\frac{3}{8}$ b) $\frac{2}{7}$ c) $\frac{2}{3}$

5102204

A pizza is cut into $12$ equal slices. Three slices are placed on a plate. a) What fraction of the whole pizza is on the plate? b) Simplify the fraction. c) Explain what the simplified fraction means if the pizza is viewed as being divided into larger equal pieces.

Hints

- Identify the total number of equal parts and the number selected. - Find a number that divides both the numerator and denominator. - Imagine removing some dividing lines to group small slices into larger equal pieces.

Solution

1. Three of the $12$ slices are on the plate, so the fraction is $\frac{3}{12}$. 2. Divide the numerator and denominator by $3$: $\frac{3}{12}=\frac{1}{4}$. 3. Grouping every $3$ small slices makes $4$ larger equal pieces. The $3$ slices on the plate make exactly $1$ of those $4$ pieces.

Answer

a) $\frac{3}{12}$ b) $\frac{1}{4}$ c) The $3$ small slices together make one-fourth of the whole pizza.

5102384

Determine whether $\frac{7}{45}$ and $\frac{14}{49}$ can be simplified. Justify each answer by identifying common factors of the numerator and denominator.

Hints

- A fraction is in simplest form when the numerator and denominator share no factor greater than $1$. - List the factors of each numerator and denominator. - Divide both parts of a fraction by the same common factor.

Solution

1. The factors of $7$ are $1$ and $7$. The factors of $45$ are $1,3,5,9,15,$ and $45$. Their only common factor is $1$, so $\frac{7}{45}$ is already in simplest form. 2. The numerator $14$ and denominator $49$ share the factor $7$. Divide both by $7$: $\frac{14}{49}=\frac{2}{7}$.

Answer

$\frac{7}{45}$ cannot be simplified. $\frac{14}{49}$ simplifies to $\frac{2}{7}$.

5118084

Which fraction can be simplified by dividing both the numerator and denominator by $6$? Then write that fraction in simplest form. a) $\frac{12}{20}$ b) $\frac{18}{30}$ c) $\frac{24}{32}$

Hints

- A number is divisible by $6$ when it is divisible by both $2$ and $3$. - Both the numerator and denominator must be divisible by the same number. - Check whether the resulting fraction can be simplified further.

Solution

1. In a), $12$ is divisible by $6$, but $20$ is not. 2. In b), both $18$ and $30$ are divisible by $6$: $\frac{18}{30}=\frac{3}{5}$. 3. In c), $24$ is divisible by $6$, but $32$ is not. 4. Therefore, only b) can be simplified by $6$, and $\frac{3}{5}$ is in simplest form.

Answer

b) $\frac{18}{30}=\frac{3}{5}$

5201704

Answer each question about fractions. a) Write the correct symbol, $<$, $>$, or $=$: $\frac{3}{10} \mathbin{\Box} \frac{7}{10}$. b) Which of these fractions are equivalent to $\frac{1}{2}$? $\frac{2}{4}, \frac{3}{8}, \frac{4}{8}, \frac{1}{3}$ c) Explain why $\frac{2}{4}$ and $\frac{4}{8}$ represent the same amount.

Hints

- For part a), compare the numerators because the denominators are the same. - A fraction equal to one-half has a numerator that is half its denominator. - For part c), imagine dividing each fourth into two equal pieces.

Solution

1. The fractions in part a) have the same denominator. Since $3 < 7$, $\frac{3}{10} < \frac{7}{10}$. 2. A fraction is equivalent to $\frac{1}{2}$ when its numerator is half its denominator. This is true for $\frac{2}{4}$ and $\frac{4}{8}$. 3. Multiplying both the numerator and denominator of $\frac{2}{4}$ by $2$ gives $\frac{4}{8}$. Dividing the same whole into twice as many equal parts requires twice as many parts to represent the same amount.

Answer

a) $\frac{3}{10} < \frac{7}{10}$ b) $\frac{2}{4}$ and $\frac{4}{8}$ c) Both fractions represent one-half of the whole; multiplying the numerator and denominator of $\frac{2}{4}$ by $2$ gives $\frac{4}{8}$.

5201714

Complete each chain of equivalent fractions by filling in the missing numbers. a) $\frac{1}{2} = \frac{2}{\Box} = \frac{\Box}{8} = \frac{5}{\Box}$ b) $\frac{1}{4} = \frac{\Box}{8}$ c) $\frac{3}{4} = \frac{6}{\Box}$ d) $\frac{10}{10} = \frac{\Box}{2}$

Hints

- Multiply or divide the numerator and denominator by the same number. - Look at how the known numerator or denominator changed. - Remember that a fraction with the same numerator and denominator equals $1$.

Solution

1. For part a), multiply the numerator and denominator by the same number: $\frac{1}{2} = \frac{2}{4} = \frac{4}{8} = \frac{5}{10}$. 2. For part b), multiply the numerator and denominator of $\frac{1}{4}$ by $2$: $\frac{1}{4} = \frac{2}{8}$. 3. For part c), the numerator is multiplied by $2$, so multiply the denominator by $2$: $\frac{3}{4} = \frac{6}{8}$. 4. For part d), $\frac{10}{10} = 1$, so the second fraction must also have equal numerator and denominator: $\frac{2}{2}$.

Answer

a) $\frac{1}{2} = \frac{2}{4} = \frac{4}{8} = \frac{5}{10}$ b) $\frac{1}{4} = \frac{2}{8}$ c) $\frac{3}{4} = \frac{6}{8}$ d) $\frac{10}{10} = \frac{2}{2}$

5202004

Two same-size rectangular pizzas are prepared for a school event. The first pizza is cut into $4$ equal pieces, and the second pizza is cut into $8$ equal pieces. a) What fraction of the first pizza is represented by $2$ pieces? b) How many pieces of the second pizza make exactly $\frac{1}{4}$ of a pizza? c) A class orders $\frac{3}{4}$ of a pizza. How many pieces from the second pizza should be packed?

Hints

- Think about how many eighth-size pieces fit in one fourth-size piece. - Rewrite fourths as eighths by multiplying the numerator and denominator by the same number. - A quick sketch of the two pizzas may help.

Solution

1. Each piece of the first pizza is $\frac{1}{4}$, so $2$ pieces represent $\frac{2}{4} = \frac{1}{2}$. 2. One fourth is equivalent to two eighths: $\frac{1}{4} = \frac{2}{8}$. Therefore, $2$ pieces of the second pizza make $\frac{1}{4}$. 3. Rewrite $\frac{3}{4}$ in eighths by multiplying the numerator and denominator by $2$: $\frac{3}{4} = \frac{6}{8}$. Therefore, $6$ pieces should be packed.

Answer

a) $\frac{2}{4}$, or $\frac{1}{2}$ b) $2$ pieces c) $6$ pieces

5209994

One dollar equals $100$ cents. a) What fraction of a dollar is $1$ cent? What fractions of a dollar are $18$ cents and $73$ cents? Write each answer as a fraction. b) Explain why $25$ cents is exactly $\frac{1}{4}$ of a dollar.

Hints

- How many cents make one whole dollar? - What does the denominator tell you about the number of equal parts in the whole? - If a dollar is divided into four equal amounts, how many cents are in each amount?

Solution

1. Because $\$1 = 100$ cents, $1$ cent is $\frac{1}{100}$ of a dollar. 2. Similarly, $18$ cents is $\frac{18}{100}$ of a dollar, and $73$ cents is $\frac{73}{100}$ of a dollar. 3. Since $100 \div 4 = 25$, $25$ cents is one of four equal parts of a dollar. Equivalently, $\frac{25}{100} = \frac{1}{4}$.

Answer

a) $1$ cent is $\frac{1}{100}$ of a dollar; $18$ cents is $\frac{18}{100}$; and $73$ cents is $\frac{73}{100}$. b) A dollar has $100$ cents, and $100 \div 4 = 25$. Therefore, $25$ cents is $\frac{1}{4}$ of a dollar.

5319694

Find the shaded fraction in each figure. Write each fraction in simplest form.

Hints

- Count all equal parts to find the denominator. - Count the shaded parts to find the numerator. - Divide the numerator and denominator by a common factor when possible.

Solution

1. In a), $6$ of $8$ equal sectors are shaded: $\frac{6}{8}=\frac{3}{4}$. 2. In b), the grid has $3\times5=15$ equal squares, and $10$ are shaded: $\frac{10}{15}=\frac{2}{3}$. 3. In c), $2$ of $5$ equal parts are shaded, so the fraction is $\frac{2}{5}$.

Answer

a) $\frac{3}{4}$ b) $\frac{2}{3}$ c) $\frac{2}{5}$

5319734

Jan has a chocolate bar made of $24$ equal pieces. The shaded pieces have already been eaten. What fraction of the whole chocolate bar remains? Write the fraction in simplest form.

Hints

- Decide whether the question asks about the eaten pieces or the remaining pieces. - Count the total number of pieces. - Count the pieces that are not shaded. - Write the remaining part as a fraction. - Simplify the fraction.

Solution

1. The bar has $4\times6=24$ pieces. 2. There are $9$ shaded pieces, so $24-9=15$ pieces remain. 3. The remaining fraction is $\frac{15}{24}=\frac{5}{8}$.

Answer

$\frac{5}{8}$

5319874

Look at figures a), b), and c). Which two figures represent the same fraction of a whole? Write that fraction in simplest form.

Hints

- For each figure, count the total number of equal parts to find the denominator. - Count the shaded parts to find the numerator. - Simplify each fraction, then compare the results.

Solution

1. In figure a), $4$ of $8$ equal parts are shaded, so the fraction is $\frac{4}{8} = \frac{1}{2}$. 2. In figure b), $6$ of $12$ equal parts are shaded, so the fraction is $\frac{6}{12} = \frac{1}{2}$. 3. In figure c), $2$ of $5$ equal parts are shaded, so the fraction is $\frac{2}{5}$. 4. Therefore, figures a) and b) represent the same fraction, $\frac{1}{2}$.

Answer

Figures a) and b) represent the same fraction. In simplest form, the fraction is $\frac{1}{2}$.

5319894

A box holds chocolates in a grid. The blue-shaded spaces still contain chocolates, and the unshaded spaces are empty. a) What fraction of all spaces are still filled? Write the fraction in simplest form. b) What fraction of all spaces are empty? Write the fraction in simplest form.

Hints

- Count all spaces in the grid. - Count the blue-shaded spaces and the unshaded spaces. - Write each count over the total and simplify.

Solution

1. The grid has $3$ rows and $5$ columns, so there are $3\times5=15$ spaces. 2. Six spaces are blue-shaded, so the filled fraction is $\frac{6}{15}=\frac{2}{5}$. 3. The number of empty spaces is $15-6=9$, so the empty fraction is $\frac{9}{15}=\frac{3}{5}$.

Answer

a) $\frac{2}{5}$ b) $\frac{3}{5}$

5319904

Ava and Luis have same-size round cakes. Ava's cake, shown in figure A, is cut into $4$ equal pieces, and she has eaten $3$ pieces. Luis's cake, shown in figure B, is cut into $8$ equal pieces. He wants to eat the same fraction of his cake as Ava ate. a) What fraction of Ava's cake has been eaten? b) How many pieces of Luis's cake must he eat? c) What equivalent fraction describes the part of Luis's cake he will eat?

Hints

- Count the total pieces and the shaded pieces in figure A. - Figure B has twice as many equal pieces as figure A. - Multiply the numerator and denominator of Ava's fraction by the same number.

Solution

1. Ava ate $3$ of the $4$ equal pieces, so she ate $\frac{3}{4}$ of her cake. 2. To write $\frac{3}{4}$ in eighths, multiply the numerator and denominator by $2$: $\frac{3 \times 2}{4 \times 2} = \frac{6}{8}$. 3. Luis must eat $6$ pieces, which is $\frac{6}{8}$ of his cake.

Answer

a) $\frac{3}{4}$ b) $6$ pieces c) $\frac{6}{8}$

5319984

A chocolate bar is divided into equal pieces, as shown. The orange-shaded pieces remain, and the unshaded pieces have been eaten. a) How many pieces were in the whole chocolate bar? b) What fraction of the chocolate bar remains? Write the fraction in simplest form. c) What fraction of the chocolate bar has been eaten? Write the fraction in simplest form.

Hints

- Multiply the number of rows by the number of columns to find the total number of pieces. - Put the number of pieces in the part over the total number of pieces. - Divide the numerator and denominator by a common factor to simplify.

Solution

1. The array has $3$ rows and $5$ columns, so the bar had $3\times5=15$ pieces. 2. There are $9$ orange-shaded pieces. The fraction remaining is $\frac{9}{15}=\frac{3}{5}$. 3. There are $15-9=6$ unshaded pieces. The fraction eaten is $\frac{6}{15}=\frac{2}{5}$.

Answer

a) $15$ pieces b) $\frac{3}{5}$ c) $\frac{2}{5}$

5319994

What fraction of the squares in the grid are blue-shaded? Write the fraction in simplest form.

Hints

- Count all squares to find the denominator. - Count the blue-shaded squares to find the numerator. - Simplify using a common factor.

Solution

1. The grid has $3$ rows and $5$ columns, so it contains $3\times5=15$ squares. 2. Nine squares are blue-shaded, so the fraction is $\frac{9}{15}$. 3. Divide the numerator and denominator by $3$: $\frac{9}{15}=\frac{3}{5}$.

Answer

$\frac{3}{5}$

5320154

Some marbles in the box are blue, as shown. What fraction of the marbles are blue? Write the fraction in simplest form.

Hints

- Count all marbles. - Count the blue marbles. - Write the blue count over the total count and simplify.

Solution

1. There are $20$ marbles in all. 2. Eight marbles are blue, so the fraction is $\frac{8}{20}$. 3. Divide the numerator and denominator by $4$: $\frac{8}{20}=\frac{2}{5}$.

Answer

$\frac{2}{5}$

5320474

Consider the grid of equal squares. a) What fraction of the grid is blue-shaded? Write the fraction in simplest form. b) What fraction of the grid is unshaded? Write the fraction in simplest form.

Hints

- Count all equal squares in the grid. - Count the blue-shaded squares. - Subtract from the total to find the unshaded count, then simplify both fractions.

Solution

1. The grid has $4$ rows and $6$ columns, so it contains $4\times6=24$ squares. 2. Nine squares are blue-shaded, so the shaded fraction is $\frac{9}{24}=\frac{3}{8}$. 3. The number of unshaded squares is $24-9=15$, so the unshaded fraction is $\frac{15}{24}=\frac{5}{8}$.

Answer

a) $\frac{3}{8}$ b) $\frac{5}{8}$

5353144

Find the fraction in simplest form represented by each point $A$, $B$, and $C$ on the number line.

Hints

- Count the equal intervals from $0$ to $1$. - What fraction does one small interval represent? - Count intervals from $0$ to determine each numerator. - Simplify each fraction completely.

Solution

1. The interval from $0$ to $1$ is divided into $6$ equal parts, so each interval represents $\frac{1}{6}$. 2. Point $A$ is at the first tick, so $A = \frac{1}{6}$. 3. Point $B$ is at the second tick, so $B = \frac{2}{6} = \frac{1}{3}$. 4. Point $C$ is at the fourth tick, so $C = \frac{4}{6} = \frac{2}{3}$.

Answer

$A = \frac{1}{6}$, $B = \frac{1}{3}$, and $C = \frac{2}{3}$.

5353154

What fractions are represented by points $P$, $Q$, and $R$? Write each answer in simplest form.

Hints

- Count the equal intervals from $0$ to $1$ to determine the denominator. - Each point represents a certain number of those intervals. - Check whether the numerator and denominator have a common factor.

Solution

1. The interval from $0$ to $1$ is divided into $8$ equal parts, so each interval represents $\frac{1}{8}$. 2. Point $P$ is at the second tick: $\frac{2}{8} = \frac{1}{4}$. 3. Point $Q$ is at the fifth tick: $\frac{5}{8}$. 4. Point $R$ is at the sixth tick: $\frac{6}{8} = \frac{3}{4}$.

Answer

$P = \frac{1}{4}$, $Q = \frac{5}{8}$, and $R = \frac{3}{4}$.

5354844

An artist is designing a wall mosaic. The blue-shaded squares represent colored glass, and the unshaded squares represent the background. What fraction of the total area is covered by blue glass? Write the fraction in simplest form.

Hints

- Count all the equal squares in the grid. - Count the blue-shaded squares. - Write the shaded count over the total count and simplify.

Solution

1. The grid has $4$ rows and $6$ columns, so it contains $4\times6=24$ equal squares. 2. There are $9$ blue-shaded squares, so the fraction is $\frac{9}{24}$. 3. Simplify by dividing by $3$: $\frac{9}{24}=\frac{3}{8}$.

Answer

The blue glass covers $\frac{3}{8}$ of the total area.

5355074

Jordan and Mia look at the circle. Jordan says, “Exactly one-half of the circle is shaded blue.” Mia says, “No, two-fourths of the circle is shaded.” Who is correct? Use the figure to explain your answer.

Hints

- Count the total number of equal parts in the circle. - Count the shaded parts and write that fraction. - Simplify the fraction and compare it with one-half.

Solution

1. The circle is divided into $4$ equal parts, and $2$ parts are shaded. The shaded fraction is $\frac{2}{4}$. 2. Simplify $\frac{2}{4}$ by dividing the numerator and denominator by $2$: $\frac{2 \div 2}{4 \div 2} = \frac{1}{2}$. 3. Since $\frac{2}{4} = \frac{1}{2}$, both students describe the same shaded amount and are correct.

Answer

Both Jordan and Mia are correct. The figure shows $\frac{2}{4}$, and $\frac{2}{4} = \frac{1}{2}$.

5355564

This hexagonal tile is divided into $6$ equal triangles. a) What fraction of the tile is shaded? b) Write an equivalent fraction with a denominator of $2$.

Hints

- Count the shaded parts and the total number of equal parts. - An equivalent fraction names the same amount with different numbers. - Look for a common factor of the numerator and denominator.

Solution

1. Three of the $6$ equal triangles are shaded, so the shaded fraction is $\frac{3}{6}$. 2. Divide the numerator and denominator by $3$: $\frac{3 \div 3}{6 \div 3} = \frac{1}{2}$.

Answer

a) $\frac{3}{6}$ b) $\frac{1}{2}$

5355594

A class survey asked students whether they enjoy playing sports in their free time. In the circle model, the green-shaded sectors represent students who said yes, and the unshaded sectors represent students who said no. a) What fraction of the students said they enjoy playing sports? Write the fraction in simplest form. b) What fraction of the students said they do not enjoy playing sports? Write the fraction in simplest form.

Hints

- Count all equal sectors to find the denominator. - Count the green-shaded sectors for part a). - Count the unshaded sectors for part b), then simplify each fraction.

Solution

1. The circle is divided into $12$ equal sectors, and $9$ are green-shaded. 2. The fraction who said yes is $\frac{9}{12}=\frac{3}{4}$. 3. The number of unshaded sectors is $12-9=3$. The fraction who said no is $\frac{3}{12}=\frac{1}{4}$.

Answer

a) $\frac{3}{4}$ b) $\frac{1}{4}$

5355854

A box contains $15$ chocolates. Some are dark chocolate, and the rest are milk chocolate. The dark chocolates are shaded gray in the figure. What fraction of the chocolates are dark chocolate? Write the fraction in simplest form.

Hints

- Count all the chocolates to find the denominator. - Count the gray chocolates to find the numerator. - Divide the numerator and denominator by the same common factor.

Solution

1. There are $15$ chocolates in all. 2. Six chocolates are shaded gray, so the fraction that are dark chocolate is $\frac{6}{15}$. 3. Divide the numerator and denominator by $3$: $\frac{6 \div 3}{15 \div 3} = \frac{2}{5}$.

Answer

The fraction of the chocolates that are dark chocolate is $\frac{2}{5}$.

5355864

Eli and Noah each ordered a same-size pizza. Eli ate $\frac{1}{3}$ of his pizza. Noah cut his pizza into $12$ equal pieces. How many pieces must Noah eat to eat the same fraction of his pizza as Eli?

Hints

- Noah's pizza has $12$ equal pieces. - Find a fraction equivalent to $\frac{1}{3}$ with denominator $12$. - What number changes $3$ into $12$?

Solution

1. Write $\frac{1}{3}$ as an equivalent fraction with denominator $12$. 2. Multiply the numerator and denominator by $4$: $\frac{1 \times 4}{3 \times 4} = \frac{4}{12}$. 3. Noah must eat $4$ pieces.

Answer

Noah must eat $4$ pieces.

5355874

A garden bed is divided into equal squares. Strawberries grow in the green-shaded squares. What fraction of the garden bed is planted with strawberries? Write the fraction in simplest form.

Hints

- Count the total number of equal squares. - Count the green-shaded squares. - Write the shaded count over the total count and simplify.

Solution

1. The grid has $4$ rows and $5$ columns, so it contains $4\times5=20$ equal squares. 2. There are $8$ green-shaded squares, so the fraction is $\frac{8}{20}$. 3. Simplify by dividing by $4$: $\frac{8}{20}=\frac{2}{5}$.

Answer

Strawberries cover $\frac{2}{5}$ of the garden bed.

5356524

An art set contains $18$ paint pots. The purple-shaded pots in the diagram have been used. What fraction of the paint pots have been used? Write the fraction in simplest form.

Hints

- Count all paint pots. - Count the purple-shaded pots. - Write the shaded count over the total and simplify.

Solution

1. There are $18$ paint pots in all. 2. Six pots are purple-shaded, so the fraction used is $\frac{6}{18}$. 3. Divide the numerator and denominator by $6$: $\frac{6}{18}=\frac{1}{3}$.

Answer

$\frac{1}{3}$

5357104

For each figure, find the shaded fraction. Write the fraction shown, then simplify it.

Hints

- Count the total number of equal parts and the number of shaded parts. - Write the shaded count as the numerator and the total count as the denominator. - Divide the numerator and denominator by a common factor.

Solution

1. In figure a), the grid has $3\times4=12$ equal squares. Nine are shaded, so the fraction is $\frac{9}{12}$. Divide the numerator and denominator by $3$: $\frac{9}{12}=\frac{3}{4}$. 2. In figure b), the hexagon has $6$ equal parts. Four are shaded, so the fraction is $\frac{4}{6}$. Divide the numerator and denominator by $2$: $\frac{4}{6}=\frac{2}{3}$.

Answer

a) $\frac{9}{12}=\frac{3}{4}$ b) $\frac{4}{6}=\frac{2}{3}$

5357114

A cake is cut into $12$ equal slices. The blue-shaded slices in the diagram are still for sale. a) What fraction of the cake has already been sold? Write the fraction in simplest form. b) How many slices have been sold?

Hints

- Count the total number of slices and the blue-shaded slices that remain. - Subtract to find the number sold. - Write the number sold over the total number of slices and simplify.

Solution

1. The cake has $12$ slices, and $8$ blue-shaded slices remain. 2. The number sold is $12-8=4$. 3. The fraction sold is $\frac{4}{12}=\frac{1}{3}$.

Answer

a) $\frac{1}{3}$ b) $4$ slices

5357214

A wall mosaic has $24$ square tiles. Eight tiles are blue, and the rest are white. What fraction of the mosaic is blue? What fraction is white? Write both fractions in simplest form.

Hints

- Count the total number of tiles and the number of blue tiles. - Write each color count over the total number of tiles. - Simplify both fractions.

Solution

1. The grid has $4$ rows and $6$ columns, so it contains $4\times6=24$ tiles. 2. Eight tiles are blue, so the blue fraction is $\frac{8}{24}=\frac{1}{3}$. 3. The number of white tiles is $24-8=16$, so the white fraction is $\frac{16}{24}=\frac{2}{3}$.

Answer

Blue: $\frac{1}{3}$ White: $\frac{2}{3}$

5358054

A hexagonal glass panel is divided into $6$ equal triangles. Some triangles are tinted, as shown. What fraction of the panel is tinted? Write the fraction in simplest form.

Hints

- Count the total number of equal triangles. - Count the tinted triangles. - Simplify the fraction by dividing the numerator and denominator by the same number.

Solution

1. The panel is divided into $6$ equal triangles. 2. Four triangles are tinted, so the tinted fraction is $\frac{4}{6}$. 3. Divide the numerator and denominator by $2$: $\frac{4 \div 2}{6 \div 2} = \frac{2}{3}$.

Answer

The tinted fraction of the panel is $\frac{2}{3}$.

5358194

A square mosaic is made of $16$ equal tiles. Some tiles are blue, as shown. What fraction of the entire mosaic is blue? Write the fraction in simplest form.

Hints

- Count all the small squares in the grid. - Count the blue squares. - Simplify the fraction by dividing the numerator and denominator by a common factor.

Solution

1. The mosaic has $4 \times 4 = 16$ tiles. 2. Six tiles are blue, so the blue fraction is $\frac{6}{16}$. 3. Divide the numerator and denominator by $2$: $\frac{6 \div 2}{16 \div 2} = \frac{3}{8}$.

Answer

The blue fraction of the mosaic is $\frac{3}{8}$.

5358454

What fraction of each figure is shaded? Write each fraction in simplest form.

Hints

- Count all equal parts or objects for the denominator. - Count the shaded parts or objects for the numerator. - Simplify by dividing the numerator and denominator by a common factor.

Solution

1. In a), $4$ of $9$ equal parts are shaded, so the fraction is $\frac{4}{9}$. 2. In b), the grid has $4\times5=20$ equal rectangles, and $12$ are shaded. Thus, $\frac{12}{20}=\frac{3}{5}$. 3. In c), $6$ of $18$ objects are shaded. Thus, $\frac{6}{18}=\frac{1}{3}$.

Answer

a) $\frac{4}{9}$ b) $\frac{3}{5}$ c) $\frac{1}{3}$

5358524

What fraction of the circle is shaded blue? Write the fraction in simplest form.

Hints

- Count all equal sectors in the circle. - Count the blue sectors. - Simplify the fraction using a common factor.

Solution

1. The circle has $10$ equal sectors, and $4$ are blue. 2. The shaded fraction is $\frac{4}{10}$. 3. Divide the numerator and denominator by $2$: $\frac{4}{10}=\frac{2}{5}$.

Answer

The blue-shaded fraction is $\frac{2}{5}$.

5358554

Find the fraction of the grid that is shaded. First write the fraction with denominator $24$, then simplify it.

Hints

- Count all squares to confirm the denominator. - Count the shaded squares for the numerator. - Divide the numerator and denominator by their greatest common factor.

Solution

1. The grid has $3\times8=24$ equal squares. 2. Eighteen squares are shaded, so the fraction is $\frac{18}{24}$. 3. Divide the numerator and denominator by $6$: $\frac{18}{24}=\frac{3}{4}$.

Answer

$\frac{18}{24}=\frac{3}{4}$

5358564

Some circles in the group are blue. What fraction of the circles are blue? Write your answer in simplest form.

Hints

- Count all the circles. - Count the blue circles. - Write the blue count over the total and simplify.

Solution

1. There are $5\times9=45$ circles in all. 2. Twenty-seven circles are blue, so the fraction is $\frac{27}{45}$. 3. Divide the numerator and denominator by $9$: $\frac{27}{45}=\frac{3}{5}$.

Answer

$\frac{3}{5}$

5374134

The blue dots in group A and the green dots in group B each represent a fraction of a group. Show with calculations that the two fractions are equivalent.

Hints

- For each group, write the fraction as shaded dots over total dots. - Simplify both fractions completely.

Solution

1. In group A, $12$ of $24$ dots are blue: $\frac{12}{24} = \frac{1}{2}$. 2. In group B, $18$ of $36$ dots are green: $\frac{18}{36} = \frac{1}{2}$. 3. Since both fractions simplify to $\frac{1}{2}$, the fractions are equivalent.

Answer

In group A, $\frac{12}{24} = \frac{1}{2}$, and in group B, $\frac{18}{36} = \frac{1}{2}$. Therefore, the fractions are equivalent.

5374374

A survey asked $50$ students how they usually travel to school. Panel a) shows the $20$ students who bike, panel b) shows the $18$ students who walk, and panel c) shows the $12$ students who ride the bus. Find the difference between the largest and smallest groups. Then find the fraction of the students who bike, in simplest form.

Hints

- Compare the three group sizes first. - Write the number who bike over the total number surveyed. - Simplify $\frac{20}{50}$ by dividing the numerator and denominator by the same factor.

Solution

1. The largest group has $20$ students, and the smallest has $12$. The difference is $20 - 12 = 8$ students. 2. The fraction who bike is $\frac{20}{50}$. Divide the numerator and denominator by $10$: $\frac{20}{50} = \frac{2}{5}$.

Answer

Difference: $8$ students; fraction who bike: $\frac{2}{5}$

5101954

Given $\frac{7}{12}=\frac{a}{84}=\frac{91}{b}$: 1) Find $a$ and $b$. 2) What scale factor $k$ changes $\frac{7}{12}$ to an equivalent fraction with denominator $132$?

Hints

- Compare each equivalent fraction with $\frac{7}{12}$. - Use the same scale factor for the numerator and denominator. - For part 2), divide the target denominator by $12$.

Solution

1. To find $a$, use $84\div12=7$. Then $a=7\times7=49$. 2. To find $b$, use $91\div7=13$. Then $b=12\times13=156$. 3. For denominator $132$, $k=132\div12=11$.

Answer

1) $a=49$, $b=156$ 2) $k=11$

5101974

Determine whether each pair of fractions is equivalent. Rewrite the fraction with the smaller denominator so both fractions have the same denominator, then compare the numerators. a) $\frac{5}{6}$ and $\frac{21}{24}$ b) $\frac{4}{7}$ and $\frac{28}{49}$

Hints

- Divide the larger denominator by the smaller denominator to find the scale factor. - Multiply the numerator and denominator by the same factor. - Fractions with the same denominator are equivalent only when their numerators are equal.

Solution

1. For a), rewrite $\frac{5}{6}$ with denominator $24$: $\frac{5\times4}{6\times4}=\frac{20}{24}$. Since $\frac{20}{24}\ne\frac{21}{24}$, the fractions are not equivalent. 2. For b), rewrite $\frac{4}{7}$ with denominator $49$: $\frac{4\times7}{7\times7}=\frac{28}{49}$. The fractions are equivalent.

Answer

a) Not equivalent, because $\frac{20}{24}\ne\frac{21}{24}$. b) Equivalent, because $\frac{4}{7}=\frac{28}{49}$.

5102004

For the fraction $\frac{126}{162}$, determine which of these numbers can divide both the numerator and denominator: $2, 3, 4, 6, 9$. List all that work.

Hints

- A number can reduce a fraction only when it divides both the numerator and denominator. - Use divisibility rules for $2$, $3$, $4$, $6$, and $9$. - Check each proposed number separately.

Solution

1. Both $126$ and $162$ are even, so $2$ works. 2. Each number has digit sum $9$, so both are divisible by $3$ and $9$. 3. Neither number is divisible by $4$. 4. Both are divisible by $2$ and $3$, so both are divisible by $6$. 5. Therefore, $2, 3, 6,$ and $9$ work.

Answer

$2, 3, 6,$ and $9$

5102014

A fraction can be simplified in one step or in several steps. Consider $\frac{210}{462}$. a) Divide the numerator and denominator first by $2$, then by $3$, and finally by $7$. What fraction remains in simplest form? b) What single number could you divide the original numerator and denominator by to reach the same result directly?

Hints

- Use each new fraction as the starting point for the next division. - Dividing in sequence by several numbers is equivalent to dividing once by their product. - Multiply the three divisors from part a).

Solution

1. Simplify step by step: $\frac{210}{462}\rightarrow\frac{105}{231}\rightarrow\frac{35}{77}\rightarrow\frac{5}{11}$. 2. The combined divisor is $2\times3\times7=42$. 3. Check: $210\div42=5$ and $462\div42=11$.

Answer

a) $\frac{5}{11}$ b) $42$

5102044

Find the missing values in each chain of equivalent fractions. 1) $\frac{4}{7}=\frac{a}{28}=\frac{36}{b}$ 2) $\frac{120}{180}=\frac{c}{15}=\frac{2}{d}$

Hints

- Compare each incomplete fraction with the first complete fraction in its chain. - Use the same scale factor for the numerator and denominator. - Verify that every fraction in a chain has the same value.

Solution

1. In the first chain, $28\div7=4$, so $a=4\times4=16$. Also, $36\div4=9$, so $b=7\times9=63$. 2. In the second chain, $180\div15=12$, so $c=120\div12=10$. Also, $120\div2=60$, so $d=180\div60=3$.

Answer

1) $a=16$, $b=63$ 2) $c=10$, $d=3$

5102094

Do $\frac{84}{105}$ and $\frac{64}{80}$ have the same value? Simplify both fractions completely and compare.

Hints

- Simplify each fraction separately. - Equivalent fractions have the same simplest form. - Find a common factor of the numerator and denominator in each fraction.

Solution

1. Divide $84$ and $105$ by $21$: $\frac{84}{105}=\frac{4}{5}$. 2. Divide $64$ and $80$ by $16$: $\frac{64}{80}=\frac{4}{5}$. 3. Both fractions simplify to $\frac{4}{5}$, so they are equivalent.

Answer

Yes. Both fractions simplify to $\frac{4}{5}$.

5102104

A fraction was simplified by dividing its numerator and denominator by a positive whole number $k$. The result was $\frac{9}{13}$, and the original numerator was $117$. Find $k$ and the original denominator.

Hints

- Divide the original numerator by the simplified numerator. - Simplifying uses the same divisor for the numerator and denominator. - Reverse the simplification by multiplying the simplified denominator by $k$.

Solution

1. Since $117\div k=9$, $k=117\div9=13$. 2. The original denominator must also be $13$ times the simplified denominator: $13\times13=169$. 3. The original fraction was $\frac{117}{169}$.

Answer

$k=13$, and the original denominator was $169$.

5102164

Three of these fractions are equivalent. Which fraction does not belong? Simplify all four fractions to justify your answer. $\frac{18}{24},\ \frac{25}{40},\ \frac{45}{60},\ \frac{9}{12}$

Hints

- Simplify each fraction completely. - Fractions with the same simplest form are equivalent. - Identify the one result that differs from the other three.

Solution

1. $\frac{18}{24}=\frac{3}{4}$. 2. $\frac{25}{40}=\frac{5}{8}$. 3. $\frac{45}{60}=\frac{3}{4}$. 4. $\frac{9}{12}=\frac{3}{4}$. 5. Therefore, $\frac{25}{40}$ does not belong because it simplifies to $\frac{5}{8}$, while the other three simplify to $\frac{3}{4}$.

Answer

$\frac{25}{40}$ does not belong.

5102174

A fraction is to be rewritten as an equivalent fraction with denominator $20$. a) List all possible original denominators other than $1$ and $20$. b) For each denominator from part a), give one example of a fraction less than $1$ that can be rewritten with denominator $20$.

Hints

- An original denominator must be a factor of $20$. - List the factor pairs of $20$. - A fraction is less than $1$ when its numerator is less than its denominator.

Solution

1. The positive factors of $20$ are $1, 2, 4, 5, 10,$ and $20$. 2. Excluding $1$ and $20$, the possible denominators are $2, 4, 5,$ and $10$. 3. Examples are $\frac{1}{2}=\frac{10}{20}$, $\frac{1}{4}=\frac{5}{20}$, $\frac{1}{5}=\frac{4}{20}$, and $\frac{1}{10}=\frac{2}{20}$.

Answer

a) $2, 4, 5,$ and $10$ b) For example, $\frac{1}{2},\frac{1}{4},\frac{1}{5},$ and $\frac{1}{10}$, respectively.

5102214

Find $a$, $x$, and $y$ in this chain of equivalent fractions: $\frac{2}{5}\xrightarrow{\text{multiply by }a}\frac{x}{20}\xrightarrow{\text{divide by }2}\frac{4}{y}$

Hints

- Find the factor that changes denominator $5$ to denominator $20$. - Apply the same factor to the numerator. - In the second step, divide both the numerator and denominator by $2$.

Solution

1. Since $5\times a=20$, $a=4$. 2. Multiply the numerator by the same factor: $x=2\times4=8$. 3. Divide the denominator $20$ by $2$: $y=10$.

Answer

$a=4$, $x=8$, $y=10$

5118074

Jan claims, “A fraction with an odd numerator and an odd denominator can never be simplified.” Which fraction is a counterexample to Jan's claim? Simplify it completely. $\frac{7}{9},\ \frac{15}{25},\ \frac{3}{11},\ \frac{17}{19}$

Hints

- A counterexample is one case that makes a general claim false. - Look for an odd numerator and odd denominator that share a common factor. - Use the divisibility rule for $5$.

Solution

1. The numerator and denominator of $\frac{15}{25}$ are both divisible by $5$. 2. Simplify: $\frac{15}{25}=\frac{3}{5}$. 3. Therefore, $\frac{15}{25}$ is a counterexample because both numbers are odd, yet the fraction can be simplified.

Answer

$\frac{15}{25}$, which simplifies to $\frac{3}{5}$

5201934

A rectangular flower bed is divided into $16$ equal squares. A gardener plants $\frac{1}{4}$ of the bed with tulips and $\frac{1}{8}$ with daffodils. All the remaining squares are planted with roses. How many squares are planted with roses? What fraction of the entire bed is planted with roses?

Hints

- How many squares are one-fourth of $16$? - How many squares are one-eighth of $16$? - After accounting for the tulips and daffodils, how many squares remain? - Write the rose squares as a fraction of all $16$ squares.

Solution

1. Find the tulip squares: $\frac{1}{4} \times 16 = 4$. 2. Find the daffodil squares: $\frac{1}{8} \times 16 = 2$. 3. Find the number of planted squares already used: $4 + 2 = 6$. 4. Find the rose squares: $16 - 6 = 10$. 5. The rose fraction is $\frac{10}{16}$, which simplifies to $\frac{5}{8}$.

Answer

There are $10$ squares planted with roses. The roses cover $\frac{5}{8}$ of the bed (or $\frac{10}{16}$).

5202014

Fill in each blank to make the fractions equivalent. Then compare the fractions in part d). a) $\frac{1}{2} = \frac{\Box}{8}$ b) $\frac{3}{4} = \frac{\Box}{8}$ c) $\frac{6}{8} = \frac{\Box}{4}$ d) Which fraction is greater, $\frac{1}{2}$ or $\frac{5}{8}$? Explain by writing both fractions with a denominator of $8$.

Hints

- Equivalent fractions can be created by multiplying or dividing the numerator and denominator by the same number. - For part d), rewrite both fractions with denominator $8$. - Once the denominators match, compare the numerators.

Solution

1. Multiply the numerator and denominator of $\frac{1}{2}$ by $4$: $\frac{1}{2} = \frac{4}{8}$. 2. Multiply the numerator and denominator of $\frac{3}{4}$ by $2$: $\frac{3}{4} = \frac{6}{8}$. 3. Divide the numerator and denominator of $\frac{6}{8}$ by $2$: $\frac{6}{8} = \frac{3}{4}$. 4. Rewrite $\frac{1}{2}$ as $\frac{4}{8}$. Since $5 > 4$, $\frac{5}{8} > \frac{4}{8}$, so $\frac{5}{8}$ is greater.

Answer

a) $\frac{1}{2} = \frac{4}{8}$ b) $\frac{3}{4} = \frac{6}{8}$ c) $\frac{6}{8} = \frac{3}{4}$ d) $\frac{5}{8}$ is greater because $\frac{1}{2} = \frac{4}{8}$ and $\frac{5}{8} > \frac{4}{8}$.

5210004

In US currency, $\$1$ equals $100$ cents. a) Write $1$ cent, $5$ cents, and $40$ cents as fractions of $\$1$. b) Lucas says, “I have $\frac{25}{100}$ of a dollar in my pocket.” Give two different ways he could have exactly this amount using pennies, nickels, dimes, and quarters.

Hints

- How many cents make one whole dollar? - What does the numerator represent in a fraction with denominator $100$? - Which US coins can be combined to make $25$ cents?

Solution

1. Since one dollar is $100$ cents, $1$ cent is $\frac{1}{100}$ of a dollar. 2. Therefore, $5$ cents is $\frac{5}{100}$ of a dollar, and $40$ cents is $\frac{40}{100}$ of a dollar. 3. The fraction $\frac{25}{100}$ of a dollar equals $25$ cents. 4. One possible coin combination is one quarter. 5. Another possible combination is two dimes and one nickel.

Answer

a) $1$ cent is $\frac{1}{100}$ of a dollar; $5$ cents is $\frac{5}{100}$; and $40$ cents is $\frac{40}{100}$. b) Possible answers include one quarter, or two dimes and one nickel. Other combinations totaling $25$ cents are also correct.

5319704

Four figures show shaded fractions. Which two figures represent the same fraction? Find the fraction shown by each figure in simplest form.

Hints

- Write the shaded count over the total number of equal parts for each figure. - Simplify every fraction. - Compare the fractions in simplest form.

Solution

1. In a), $4$ of $6$ equal sectors are shaded: $\frac{4}{6}=\frac{2}{3}$. 2. In b), $12$ of $16$ equal squares are shaded: $\frac{12}{16}=\frac{3}{4}$. 3. In c), $6$ of $9$ equal parts are shaded: $\frac{6}{9}=\frac{2}{3}$. 4. In d), $2$ of $4$ equal parts are shaded: $\frac{2}{4}=\frac{1}{2}$. 5. Figures a) and c) both represent $\frac{2}{3}$.

Answer

a) $\frac{2}{3}$ b) $\frac{3}{4}$ c) $\frac{2}{3}$ d) $\frac{1}{2}$ Figures a) and c) match.

5355774

In which groups is exactly one-fourth, $\frac{1}{4}$, of the objects shaded? List the letters of all correct groups.

Hints

- Count the total number of objects in each group. - Count the shaded objects and write a fraction for each group. - Simplify each fraction and check whether it equals $\frac{1}{4}$.

Solution

1. In group a), $2$ of $8$ objects are shaded: $\frac{2}{8} = \frac{1}{4}$. 2. In group b), $2$ of $6$ objects are shaded: $\frac{2}{6} = \frac{1}{3}$. 3. In group c), $3$ of $12$ objects are shaded: $\frac{3}{12} = \frac{1}{4}$. 4. In group d), $2$ of $10$ objects are shaded: $\frac{2}{10} = \frac{1}{5}$. 5. Therefore, groups a) and c) show one-fourth shaded.

Answer

Groups a) and c).

5357844

For each figure, write the purple-shaded part as a fraction and simplify it. What do you notice when you compare the two results?

Hints

- For each figure, count the shaded parts and the total equal parts. - Simplify each fraction by dividing the numerator and denominator by a common factor. - Compare the simplified fractions.

Solution

1. In figure a), $6$ of the $8$ equal parts are shaded, so the fraction is $\frac{6}{8}=\frac{3}{4}$. 2. In figure b), the grid has $3\times4=12$ equal squares. Nine are shaded, so the fraction is $\frac{9}{12}=\frac{3}{4}$. 3. Both fractions simplify to $\frac{3}{4}$, so they are equivalent and represent the same portion of a whole.

Answer

a) $\frac{6}{8}=\frac{3}{4}$ b) $\frac{9}{12}=\frac{3}{4}$ Both figures show the same fraction of a whole.

5358704

Look at figures a), b), and c). a) Write the fraction of each figure that is blue. b) Two figures show the same fraction of the whole even though they are divided differently. Which figures are they? Explain by simplifying or generating equivalent fractions.

Hints

- Count the equal parts and the blue parts in each figure. - Simplify each fraction. - Compare the simplified fractions rather than the number of dividing lines.

Solution

1. Figure a) has $1$ of $4$ equal parts shaded, so it represents $\frac{1}{4}$. 2. Figure b) has $2$ of $8$ equal parts shaded, so it represents $\frac{2}{8}$. 3. Figure c) has $2$ of $6$ equal parts shaded, so it represents $\frac{2}{6}$. 4. Simplify $\frac{2}{8}$ by dividing the numerator and denominator by $2$: $\frac{2}{8} = \frac{1}{4}$. Therefore, figures a) and b) show the same fraction. Figure c) represents $\frac{2}{6} = \frac{1}{3}$.

Answer

a) a): $\frac{1}{4}$; b): $\frac{2}{8}$; c): $\frac{2}{6}$ b) Figures a) and b) show the same fraction because $\frac{2}{8} = \frac{1}{4}$.

5374114

Of $60$ dots, $36$ are shaded blue. What fraction of all the dots is blue? Simplify the fraction completely, and also find the unshaded fraction.

Hints

- Write the number of marked dots over the total number of dots. - Divide the numerator and denominator by a common factor.

Solution

1. The blue fraction is $\frac{36}{60}$. 2. Divide the numerator and denominator by $12$: $\frac{36}{60} = \frac{3}{5}$. 3. There are $60 - 36 = 24$ unshaded dots, so the unshaded fraction is $\frac{24}{60} = \frac{2}{5}$.

Answer

Blue: $\frac{3}{5}$; unshaded: $\frac{2}{5}$.

5102224

Two students simplify $\frac{24}{36}$. Leon divides first by $2$, then by $2$, and finally by $3$. Sophie divides by $12$ in one step. a) Find each student's result. b) Explain why both methods lead to the same result. c) State one advantage of each method.

Hints

- Carry out both simplification methods. - Multiply Leon's three divisors. - Compare finding one large common factor with finding several small common factors.

Solution

1. Leon's method gives $\frac{24}{36}\rightarrow\frac{12}{18}\rightarrow\frac{6}{9}\rightarrow\frac{2}{3}$. 2. Sophie's method gives $\frac{24}{36}\rightarrow\frac{2}{3}$ by dividing both numbers by $12$. 3. Leon's divisors have product $2\times2\times3=12$, so his repeated divisions are equivalent to Sophie's single division. 4. Sophie's method is faster when the greatest common factor is easy to identify. Leon's method can be easier when only small common factors are obvious.

Answer

a) Both students get $\frac{2}{3}$. b) $2\times2\times3=12$, so the total division is the same. c) Sophie's method is faster; Leon's method can be easier when the greatest common factor is not immediately clear.

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