031. Introducing Ratios and Simplifying

Learning Intentions

• To understand that a ratio compares two or more related quantities in a given order
• write a ratio from a description
• Use a common factor to Simplify a ratio

Pre-requisite Summary

• Understand that quantities can be compared Use numbers
• Be able to count and Identify different groups in a collection
• Know that order matters when writing comparisons
• Be able to write numbers using colons, e.g.
• Be able to Solve common factors of two or more numbers
• Understand that dividing numbers by the same non-zero whole number can keep a comparison equivalent

Worked Examples

Worked Example 1

a) Explain what a ratio compares.

b) Write the ratio of red counters to blue counters.

c) Explain why the order of the ratio matters.

Worked Example 2

Write each ratio from the description:

a) boys to girls

b) cats to dogs

c) cups of juice to cups of water

Worked Example 3

Write each ratio from the description:

a) apples, bananas and pears

b) white tiles to black tiles

Worked Example 4

Simplify each ratio using a common factor:

a)

b)

c)

Worked Example 5

Simplify each ratio using a common factor:

a)

b)

c)

Worked Example 6

A basket contains oranges and apples.

a) Write the ratio of oranges to apples.

b) Simplify the ratio.

c) Write the ratio of apples to oranges.

Problems

Problem 1

a) Explain what a ratio compares.

b) Write the ratio of green counters to yellow counters.

c) Explain why the order of the ratio matters.

Problem 2

Write each ratio from the description:

a) boys to girls

b) cats to dogs

c) cups of milk to cups of cereal

Problem 3

Write each ratio from the description:

a) apples, bananas and pears

b) white tiles to black tiles

Problem 4

Simplify each ratio using a common factor:

a)

b)

c)

Problem 5

Simplify each ratio using a common factor:

a)

b)

c)

Problem 6

A basket contains oranges and apples.

a) Write the ratio of oranges to apples.

b) Simplify the ratio.

c) Write the ratio of apples to oranges.

Exercises

Understanding and Fluency

Exercise 1.

Write each ratio from the description:

a) red marbles to blue marbles

b) boys to girls

c) spoons of sugar to cups of flour

Exercise 2.

Write each ratio from the description:

a) white counters to black counters

b) cats to dog

c) pencils to erasers

Exercise 3.

Write the ratio in the order given:

a) apples to oranges if there are apples and oranges

b) girls to boys if there are girls and boys

c) red tiles to blue tiles if there are red and blue

Exercise 4.

Simplify each ratio:

a)

b)

c)

Exercise 5.

Simplify each ratio:

a)

b)

c)

Exercise 6.

Simplify each ratio:

a)

b)

c)

Exercise 7.

Write and simplify the ratio:

a) red counters to blue counters

b) books to magazines

c) dogs to cats

Exercise 8.

Write and simplify the ratio:

a) boys to girls

b) apples to oranges

c) cups of water to cups of juice

Reasoning

Exercise 9.

Explain why the ratio is different from .

Exercise 10.

A student writes the ratio of cats to dogs as . Explain the mistake.

Exercise 11.

Explain why dividing both parts of a ratio by the same common factor gives an equivalent ratio.

Exercise 12.

A student simplifies to . Explain why this is not correct.

Problem-solving

Exercise 13.

A class has boys and girls. Write the ratio of boys to girls and simplify it.

Exercise 14.

A fruit bowl contains apples, oranges and pears. Write the ratio apples : oranges : pears and simplify it.

Exercise 15.

A recipe uses cups of flour and cups of milk. Write and simplify the ratio of flour to milk.

Exercise 16.

In a car park there are red cars and blue cars. Write the ratio of red cars to blue cars and simplify it.

Exercise 17.

A packet contains green beads and yellow beads. Write the ratio of green to yellow and simplify it.

Exercise 18.

A design uses white tiles and black tiles. Write the ratio of white to black and simplify it.

Potential Misunderstandings

• Students may think a ratio compares only one quantity rather than two or more related quantities
• Students may reverse the order of the quantities when writing a ratio
• Students may think and represent the same comparison
• Students may confuse ratios with fractions and ignore the stated order
• Students may simplify only one part of a ratio instead of all parts
• Students may divide by different numbers when simplifying a ratio
• Students may not use the greatest common factor, so the ratio may not be fully simplified
• Students may think a simplified ratio changes the actual comparison rather than keeping it equivalent

Next: 032. Ratios, Fractions and Proportions