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 Use the symbol
• Understand that equivalent ratios represent the same comparison
• Recall multiplication and division facts
• Be able to Solve 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) , multiply by

b) , divide by

c) , multiply by

Worked Example 2

State whether each ratio is simplified, and if not, simplify it:

a)

b)

c)

Worked Example 3

Simplify each ratio by dividing by the highest common factor:

a)

b)

c)

Worked Example 4

Simplify each ratio involving fractions:

a)

b)

c)

Worked Example 5

Simplify each ratio involving quantities by converting units first:

a)

b)

c)

Worked Example 6

Write two equivalent ratios, then give the simplified ratio:

a)

b)

c)

Problems

Problem 1

Write equivalent ratios for each by multiplying or dividing both parts by the given number:

a) , multiply by

b) , divide by

c) , multiply by

Problem 2

State whether each ratio is simplified, and if not, simplify it:

a)

b)

c)

Problem 3

Simplify each ratio by dividing by the highest common factor:

a)

b)

c)

Problem 4

Simplify each ratio involving fractions:

a)

b)

c)

Problem 5

Simplify each ratio involving quantities by converting units first:

a)

b)

c)

Problem 6

Write two equivalent ratios, then give the simplified ratio:

a)

b)

c)

Exercises

Understanding and Fluency

Exercise 1.

Write two equivalent ratios for each:

a)

b)

c)

Exercise 2.

Write an equivalent ratio by following the instruction:

a) , multiply by

b) , divide by

c) , multiply by

d) , divide by

Exercise 3.

State whether each ratio is already simplified:

a)

b)

c)

d)

Exercise 4.

Simplify each ratio involving whole numbers:

a)

b)

c)

d)

Exercise 5.

Simplify each ratio involving whole numbers:

a)

b)

c)

d)

Exercise 6.

Simplify each ratio involving fractions:

a)

b)

c)

d)

Exercise 7.

Simplify each ratio involving quantities by converting units if necessary:

a)

b)

c)

d)

Exercise 8.

Write the simplified ratio for each pair of quantities:

a)

b)

c)

d)

Reasoning

Exercise 9.

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

Exercise 10.

A student says that the simplified form of is because both numbers are smaller. Explain the mistake.

Exercise 11.

Noah says that and are different ratios because the numbers are different. Is he correct? Explain.

Exercise 12.

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

Exercise 13.

A student simplifies to by looking only at the numerators. Describe the error.

Problem-solving

Exercise 14.

A recipe uses flour and sugar in the ratio . Write this ratio in simplest form.

Exercise 15.

A map shows the ratio of blue counters to red counters as . Write two equivalent ratios and the simplified ratio.

Exercise 16.

A ribbon is cut into lengths of and . Write the ratio of the lengths in simplest form.

Exercise 17.

A drink mix uses juice and water in the ratio . Simplify the ratio.

Exercise 18.

A class has boys and girls. Write the ratio of boys to girls in simplest form.

Exercise 19.

A container holds of oil and 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

Next: 144. Dividing Quantities in a Ratio