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 using 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. $3:4$
• Be able to find 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 $3$ red counters to $5$ blue counters.
c) Explain why the order of the ratio matters.

Worked Example 2

Write each ratio from the description:
a) $7$ boys to $9$ girls
b) $4$ cats to $6$ dogs
c) $2$ cups of juice to $3$ cups of water

Worked Example 3

Write each ratio from the description:
a) $5$ apples, $3$ bananas and $2$ pears
b) $8$ white tiles to $4$ black tiles

Worked Example 4

Simplify each ratio using a common factor:
a) $6:8$
b) $12:18$
c) $15:20$

Worked Example 5

Simplify each ratio using a common factor:
a) $9:12$
b) $14:21$
c) $18:24:30$

Worked Example 6

A basket contains $10$ oranges and $15$ 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 $4$ green counters to $7$ yellow counters.
c) Explain why the order of the ratio matters.

Problem 2

Write each ratio from the description:
a) $6$ boys to $8$ girls
b) $5$ cats to $7$ dogs
c) $3$ cups of milk to $4$ cups of cereal

Problem 3

Write each ratio from the description:
a) $4$ apples, $2$ bananas and $6$ pears
b) $10$ white tiles to $5$ black tiles

Problem 4

Simplify each ratio using a common factor:
a) $8:12$
b) $10:15$
c) $16:24$

Problem 5

Simplify each ratio using a common factor:
a) $12:16$
b) $18:27$
c) $24:30:36$

Problem 6

A basket contains $12$ oranges and $18$ 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

1. Write each ratio from the description:
   a) $2$ red marbles to $5$ blue marbles
   b) $7$ boys to $4$ girls
   c) $3$ spoons of sugar to $2$ cups of flour

2. Write each ratio from the description:
   a) $6$ white counters to $9$ black counters
   b) $4$ cats to $1$ dog
   c) $8$ pencils to $3$ erasers

3. Write the ratio in the order given:
   a) apples to oranges if there are $5$ apples and $2$ oranges
   b) girls to boys if there are $9$ girls and $7$ boys
   c) red tiles to blue tiles if there are $4$ red and $6$ blue

4. Simplify each ratio:
   a) $4:6$
   b) $8:12$
   c) $10:15$

5. Simplify each ratio:
   a) $14:21$
   b) $16:24$
   c) $18:30$

6. Simplify each ratio:
   a) $12:18:24$
   b) $9:15:21$
   c) $20:25:30$

7. Write and simplify the ratio:
   a) $8$ red counters to $12$ blue counters
   b) $15$ books to $10$ magazines
   c) $6$ dogs to $9$ cats

8. Write and simplify the ratio:
   a) $18$ boys to $24$ girls
   b) $20$ apples to $30$ oranges
   c) $12$ cups of water to $16$ cups of juice

Reasoning

9. Explain why the ratio $3:5$ is different from $5:3$.

10. A student writes the ratio of $4$ cats to $7$ dogs as $7:4$. Explain the mistake.

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

12. A student simplifies $12:18$ to $6:12$. Explain why this is not correct.

Problem-solving

13. A class has $12$ boys and $18$ girls. Write the ratio of boys to girls and simplify it.

14. A fruit bowl contains $8$ apples, $12$ oranges and $4$ pears. Write the ratio apples : oranges : pears and simplify it.

15. A recipe uses $6$ cups of flour and $9$ cups of milk. Write and simplify the ratio of flour to milk.

16. In a car park there are $15$ red cars and $25$ blue cars. Write the ratio of red cars to blue cars and simplify it.

17. A packet contains $18$ green beads and $27$ yellow beads. Write the ratio of green to yellow and simplify it.

18. A design uses $24$ white tiles and $36$ 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 $a:b$ and $b:a$ 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