Ratios

Ratios exercise

• Determine the ratio that corresponds to each tape diagram.

  Hints

  Ratios can be represented in three different ways and can be simplified:

  • $4$ to $6$ equals $2$ to $3$
  • $4:6$ equals $2:3$
  • $\frac{4}{6}=\frac{2}{3}$

  Remember, in a ratio order matters!

  The ratio $1:3$ matches the tape diagram in the image.

  The ratio $8$ to $4$ equals $2$ to $1$, and matches the tape diagram in the image.

  Solution

  Ratios can be written in three different ways: $1$ to $3$; $1:3$; and $\frac{1}{3}$. They can also be visually represented with tables and tape diagrams.

  Remember that with ratios, order matters! $1:3$ is very different from $3:1$.

  1. The ratio $4$ to $5$ matches the tape diagram with $4$ blocks on top and $5$ blocks on the bottom.

  2. The ratio $\frac{3}{4}$ matches the tape diagram with $3$ blocks on top and $4$ blocks on the bottom.

  3. The ratio $3:6$ simplifies to $1:2$, and matches the tape diagram with $1$ block on top and $2$ blocks on the bottom.

  4. The ratio $5$ to $3$ matches the tape diagram with $5$ blocks on top and $3$ blocks on the bottom.

  5. The ratio $\frac{6}{4}$ simplifies to $\frac{3}{2}$, and matches the tape diagram with $3$ blocks on top and $2$ blocks on the bottom.

  6. The ratio $5:2$ matches the tape diagram with $5$ blocks on top and $2$ blocks on the bottom.

• Identify which ratios correspond to each table.

  Hints

  Ratios can be represented in three different ways and can be simplified:

  • $4$ to $6$ equals $2$ to $3$
  • $4:6$ equals $2:3$
  • $\frac{4}{6}=\frac{2}{3}$

  All of the ratios are read "$4$ to $6$".

  Remember that with ratios, order matters! The ratio of $2$ to $3$ is very different from $3$ to $2$.

  In the table, the ratios $9:12$ and $6:8$ both represent the smallest ratio $3:4$ for the reasons represented below:

  • If you divide both sides of the ratio $9:12$ by the GCF $3$, it simplifies to $3:4$.
  • If you if you divide both sides of the ratio $6:8$ by the GCF $2$, it also simplifies to $3:4$.

  Solution

  Table 1

  All ratios that simplify to $2$ to $5$.

  • The GCF for $6$ and $15$ is $3$. Divide both parts of the ratio $6:15$ by $3$, then $6:15$ equals $2:5$.
  • The GCF for $8$ and $20$ is $4$. Divide both parts of the ratio $8$ to $20$ by $4$, then $8$ to $20$ equals $2$ to $5$.
  • The ratio $2:5$ is another way to write $2$ to $5$.
  • The GCF for $10$ and $25$ is $5$. Divide both parts of the ratio by $5$ then $\frac{10}{25}=\frac{2}{5}$.

  Table 2

  All ratios that simplify to $1$ to $2$.

  • The GCF for $4$ and $8$ is $4$. Divide both parts of the ratio $4$ to $8$ by $4$, then $4$ to $8$ equals $1$ to $2$.
  • The GCF for $2$ and $4$ is $2$. Divide both parts of the ratio $2:4$ by $2$, then $2:4$ equals $1:2$.
  • The GCF for $10$ and $20$ is $10$. Divide both parts of the ratio $\frac{10}{20}$ by $10$, then $\frac{10}{20}=\frac{1}{2}$.
  • The GCF for $9$ and $18$ is $9$. Divide both parts of the ratio $9:18$ by $9$, then $9:18$ equals $1:2$.

  Table 3

  All ratios that simplify to $3$ to $2$.

  • The GCF for $9$ and $6$ is $3$. Divide both parts of the ratio $\frac{9}{6}$ by $3$, then $\frac{9}{6}=\frac{3}{2}$.
  • The GCF for $12$ and $18$ is $4$. Divide both parts of the ratio $12$ to $8$ by $4$, then $12$ to $8$ equals $3$ to $2$.
  • The ratio $\frac{3}{2}$ is another way to write $3$ to $2$.
  • The GCF for $15$ and $10$ is $5$. Divide both parts of the ratio $15:10$ by $5$, then $15:10$ equals $3:2$.

• Complete the table for each ratio.

  Hints

  In a ratio, order matters. This means that the ratio $4:5$ is not the same as the ratio $5:4$.

  To find an equivalent ratio, multiply both parts of the original ratio by the number of servings. For example, if 1 serving is the ratio $1:2$, then 3 servings would be the ratio $3:6$.

  Ginger Berry Smoothie

  • $1$ serving is $1$ ginger to $3$ strawberries
  • $2$ servings: $2$ to $6$
  • $4$ servings: $4:12$

  Solution

  To complete the tables, find the ratios by multiplying the original ratio by the number of servings.

  It is important to remember that in a ratio, order matters. This means that the ratio $4:5$ is not the same as the ratio $5:4$.

  1. Berry Blast Smoothie

  • $1$ serving is $5$ raspberries to $3$ blueberries
  • $2$ servings (Multiply both numbers by $2$): $10$ to $6$
  • $5$ servings (Multiply both numbers by $5$): $25:15$

  2. Hawaiian Fusion Smoothie

  • $1$ serving is $2$ pineapples to $4$ mangos
  • $3$ servings (Multiply both numbers by $3$): $6$ to $12$
  • $7$ servings (Multiply both numbers by $7$): $14:28$

  3. Green Giant Smoothie

  • $1$ serving is $6$ kale to $4$ spinach
  • $4$ servings (Multiply both numbers by $4$): $24$ to $16$
  • $5$ servings (Multiply both numbers by $5$): $30:20$

• Create the tape diagram for each ratio.

  Hints

  Both sides of the ratio $25:100$ can be divided by their greatest common factor, $25$, and simplify to, $1:4$.

  The top row of the tape diagram represents the left side of the ratio (or numerator), and the bottom row of the tape diagram represents the right side of the ratio (or denominator).

  The tape diagram represents the ratio of $\frac{25}{75}$.

  Solution

  1. If you divide both sides of the ratio, $11$ to $44$, by the GCF, $11$, the simplified ratio is $1$ to $4$. The tape diagram shows $1$ yellow square on top and $4$ yellow squares on the bottom.

  2. If you divide both sides of the ratio, $14:21$, by the GCF, $7$, the simplified ratio is $2:3$. The tape diagram shows $2$ yellow squares on top and $3$ yellow squares on the bottom.

  3. If you divide both sides of the ratio, $\frac{60}{20}$, by the GCF, $20$, the simplified ratio is $\frac{3}{1}$. The tape diagram shows $3$ yellow squares on top and $1$ yellow square on the bottom.

• Select the true statements about ratios.

  Hints

  Ratios can be represented in three different ways:

  • $1$ to $3$
  • $1:3$
  • $\frac{1}{3}$

  The ratio $2:4$ is equivalent to the ratio $1:2$ because $\frac{2}{4}=\frac{1}{2}$.

  The ratio $3$ to $1$ is very different from the ratio $1$ to $3$.

  Solution

  True Statements

  • The ratio $\frac{1}{3}$ can be represented as $1:3$.
  • A ratio is a relationship between two amounts.
  • The ratio, $\frac{8}{18}=\frac{4}{9}$.
  • Tape diagrams visually represent ratios.
  • $3$ cups of sugar for every $2$ cups of flour is equivalent to $6$ cups of sugar for every $4$ cups of flour.
  • In a ratio, order matters.

  False Statements

  The ratio, $12:9$, is equal to $3:4$.

  • This statement is false because dividing the ratio $12:9$ by $3$ results in a ratio of $4:3$. Another way to view the ratio is to set it up as a fraction, $\frac{12}{9}=\frac{4}{3}$. Remember, in a ratio order matters!

  The ratio, $4:8$ can also be represented as $\frac{2}{1}$.

  • The statement is false because the ratio $4:8$ represented as a fraction is $\frac{4}{8}=\frac{1}{2}$, not $\frac{2}{1}$. Remember, in a ratio order matters!

  The ratio $3:4$ can be expressed as $3\times 4$.

  • The ratio $3:4$ can be represented as $3$ to $4$ or $\frac{3}{4}$, not $3 \times 4$.

• Determine which table goes with which tape diagram.

  Hints

  Set up one row of numbers as a fraction, and reduce:

  • $\frac{12}{20}=\frac{3}{5}$

  This means that all of the ratios in the table reduce to $\frac{3}{5}$.

  The tape diagram represents the ratio $3:5$.

  Remember that with ratios, order matters!

  Solution

  • The ratio of peanuts to almonds is $1:3$. You can find this by simplifying any of the pairs of numbers in the table. The first line is $\frac{3}{9}=\frac{1}{3}$. Therefore, the tape diagram has $1$ block on top and $3$ blocks on the bottom.

  • The ratio of green to red is $4:7$. You can find this by simplifying any of the pairs of numbers in the table. The first line is $\frac{8}{14}=\frac{4}{7}$. Therefore, the tape diagram has $4$ blocks on top and $7$ blocks on the bottom.

  • The ratio of oranges to apples is $5:3$. You can find this by simplifying any of the pairs of numbers in the table. The first line is $\frac{10}{6}=\frac{5}{3}$. Therefore, the tape diagram has $5$ blocks on top and $3$ blocks on the bottom.

  • The ratio of girls to boys is $3:4$. You can find this by simplifying any of the pairs of numbers in the table. The first line is $\frac{6}{8}=\frac{3}{4}$. Therefore, the tape diagram has $3$ blocks on top and $4$ blocks on the bottom.

  • The ratio of tomatoes to carrots is $1:2$ or any multiple of the ratio $1:2$. Therefore, the tape diagram has $5$ blocks on top and $10$ blocks on the bottom.