143. Equivalent and Simplified Ratios

Learning Intentions

• write equivalent ratios for a given ratio by multiplying or dividing each quantity by a given number
• To understand that simplifying a ratio involves finding an equivalent ratio with no common factors
• simplify a ratio involving whole numbers by dividing by the highest common factor
• simplify ratios involving fractions
• write simplified ratios involving quantities by converting units if necessary

Pre-requisite Summary

• Know that a ratio compares two or more quantities
• Be able to write ratios using the $:$ symbol
• Understand that equivalent ratios represent the same comparison
• Recall multiplication and division facts
• Be able to find the highest common factor of whole numbers
• Be able to simplify fractions
• Be able to convert between common units such as cm and m, or g and kg

Worked Examples

Worked Example 1

Write equivalent ratios for each by multiplying or dividing both parts by the given number:
a) $3:5$, multiply by $2$
b) $12:18$, divide by $3$
c) $7:4$, multiply by $5$

Worked Example 2

State whether each ratio is simplified, and if not, simplify it:
a) $8:12$
b) $15:25$
c) $9:10$

Worked Example 3

Simplify each ratio by dividing by the highest common factor:
a) $18:24$
b) $35:49$
c) $42:56$

Worked Example 4

Simplify each ratio involving fractions:
a) ${\displaystyle \frac{1}{2}}:{\displaystyle \frac{3}{4}}$
b) ${\displaystyle \frac{2}{3}}:{\displaystyle \frac{5}{6}}$
c) ${\displaystyle \frac{3}{5}}:{\displaystyle \frac{9}{10}}$

Worked Example 5

Simplify each ratio involving quantities by converting units first:
a) $2\text{ m}:50\text{ cm}$
b) $750\text{ g}:1.5\text{ kg}$
c) $3\text{ L}:250\text{ mL}$

Worked Example 6

Write two equivalent ratios, then give the simplified ratio:
a) $14:21$
b) ${\displaystyle \frac{3}{4}}:{\displaystyle \frac{1}{8}}$
c) $1.2\text{ m}:80\text{ cm}$

Problems

Problem 1

Write equivalent ratios for each by multiplying or dividing both parts by the given number:
a) $4:7$, multiply by $3$
b) $20:30$, divide by $5$
c) $9:2$, multiply by $4$

Problem 2

State whether each ratio is simplified, and if not, simplify it:
a) $10:15$
b) $16:20$
c) $7:9$

Problem 3

Simplify each ratio by dividing by the highest common factor:
a) $24:36$
b) $28:42$
c) $63:81$

Problem 4

Simplify each ratio involving fractions:
a) ${\displaystyle \frac{1}{3}}:{\displaystyle \frac{2}{5}}$
b) ${\displaystyle \frac{3}{4}}:{\displaystyle \frac{1}{2}}$
c) ${\displaystyle \frac{5}{6}}:{\displaystyle \frac{10}{9}}$

Problem 5

Simplify each ratio involving quantities by converting units first:
a) $4\text{ m}:60\text{ cm}$
b) $500\text{ g}:2\text{ kg}$
c) $1.5\text{ L}:300\text{ mL}$

Problem 6

Write two equivalent ratios, then give the simplified ratio:
a) $18:27$
b) ${\displaystyle \frac{2}{3}}:{\displaystyle \frac{1}{9}}$
c) $2.5\text{ m}:50\text{ cm}$

Exercises

Understanding and Fluency

1. Write two equivalent ratios for each:
   a) $2:3$
   b) $5:8$
   c) $6:7$

2. Write an equivalent ratio by following the instruction:
   a) $4:9$, multiply by $2$
   b) $15:20$, divide by $5$
   c) $7:11$, multiply by $3$
   d) $24:36$, divide by $6$

3. State whether each ratio is already simplified:
   a) $3:5$
   b) $8:20$
   c) $14:21$
   d) $11:13$

4. Simplify each ratio involving whole numbers:
   a) $12:18$
   b) $20:30$
   c) $27:45$
   d) $54:72$

5. Simplify each ratio involving whole numbers:
   a) $32:40$
   b) $45:60$
   c) $49:63$
   d) $84:126$

6. Simplify each ratio involving fractions:
   a) ${\displaystyle \frac{1}{2}}:{\displaystyle \frac{1}{3}}$
   b) ${\displaystyle \frac{3}{4}}:{\displaystyle \frac{5}{8}}$
   c) ${\displaystyle \frac{2}{5}}:{\displaystyle \frac{3}{10}}$
   d) ${\displaystyle \frac{7}{9}}:{\displaystyle \frac{14}{27}}$

7. Simplify each ratio involving quantities by converting units if necessary:
   a) $3\text{ m}:90\text{ cm}$
   b) $2\text{ kg}:500\text{ g}$
   c) $2.5\text{ L}:750\text{ mL}$
   d) $1.2\text{ m}:40\text{ cm}$

8. Write the simplified ratio for each pair of quantities:
   a) $150\text{ cm}:2\text{ m}$
   b) $600\text{ mL}:1.8\text{ L}$
   c) $750\text{ g}:250\text{ g}$
   d) ${\displaystyle \frac{3}{4}}:{\displaystyle \frac{9}{8}}$

Reasoning

9. Explain why multiplying both parts of a ratio by the same number gives an equivalent ratio.

10. A student says that the simplified form of $12:18$ is $10:16$ because both numbers are smaller. Explain the mistake.

11. Noah says that $4:6$ and $2:3$ are different ratios because the numbers are different. Is he correct? Explain.

12. Explain why ratios involving quantities sometimes need unit conversion before they can be simplified.

13. A student simplifies ${\displaystyle \frac{1}{2}}:{\displaystyle \frac{3}{4}}$ to $1:3$ by looking only at the numerators. Describe the error.

Problem-solving

14. A recipe uses flour and sugar in the ratio $200\text{ g}:300\text{ g}$. Write this ratio in simplest form.

15. A map shows the ratio of blue counters to red counters as $18:24$. Write two equivalent ratios and the simplified ratio.

16. A ribbon is cut into lengths of $1.5\text{ m}$ and $75\text{ cm}$. Write the ratio of the lengths in simplest form.

17. A drink mix uses juice and water in the ratio ${\displaystyle \frac{2}{3}}:{\displaystyle \frac{5}{6}}$. Simplify the ratio.

18. A class has $14$ boys and $21$ girls. Write the ratio of boys to girls in simplest form.

19. A container holds $500\text{ mL}$ of oil and $1.5\text{ L}$ of water. Write the ratio of oil to water in simplest form.

Potential Misunderstandings

• Students may multiply or divide only one part of a ratio when finding an equivalent ratio
• Students may think equivalent ratios must use exactly the same numbers
• Students may simplify a ratio by subtracting instead of dividing
• Students may not use the highest common factor when simplifying
• Students may forget that a ratio is only fully simplified when there are no common factors
• Students may simplify fractions inside a ratio incorrectly
• Students may compare numerators only or denominators only in fractional ratios
• Students may try to simplify quantities without converting them to the same unit first
• Students may reverse the order of the quantities when rewriting the ratio