Lesson

Ratios tell us about the relative sizes of two or more values. They are often used in everyday life, whether it's for dividing up money, mixing paint colours, cooking, or mixing cement. Last lesson we looked at creating equivalent ratios and we now want to use this to find an unknown quantity in a given ratio.

For example, if a recipe called for the ratio of milk to flour to be $3:4$3:4, if I used six cups of milk, how much flour should I use? What if I instead used five cups of milk, how much flour should I use now? Sometimes we can quickly identify the equivalent ratio and sometimes it may require a few steps to ensure we obtain the correct value.

When we say that $2$2 quantities in a ratio are in proportion, we mean that they are equal, or equivalent.

The following ratios are in proportion

$3:3$3:3

$\frac{1}{2}:\frac{1}{2}$12:12

$\frac{1}{2}:\frac{3}{6}$12:36

If the following two quantities are in proportion. Find the missing value. $\frac{1}{2}:\frac{\editable{}}{4}$12:4.

**Think: **The fraction one-half is in the same proportion as what other fraction with a denominator of $4$4? We can think of this as finding an equivalent fraction, or how many quarters are equivalent to one-half?

**Do: **The answer is $2$2. We could also multiply both sides of the ratio by $4$4 to find the unknown value.

The ratio of sultanas to nuts in a brand of trail mix is always $32:56$32:56. If a snack size pack has $12$12 sultanas, how many nuts are there?

**Think: **Since $12$12 is not a factor of $32$32, we can first turn $32:56$32:56 into a simplified ratio, then find the equivalent ratio that looks like $12:\editable{?}$12:?.

**Do:** We can simplify the ratio $32:56$32:56 by decreasing both sides of the ratio until the highest common factor of the two numbers is $1$1. Use the fact that both $32$32 and $56$56 are divisible by $8$8.

$32$32 | $:$: | $56$56 | |

$\frac{32}{8}$328 | $:$: | $\frac{56}{8}$568 |
Divide both sides by $8$8. |

$4$4 | $:$: | $7$7 |

Since $4$4 and $7$7 have no common factors except for $1$1, this is a simplified ratio. So we now know that for every $4$4 sultanas there are $7$7 nuts.

To get from $4$4 sultanas to $12$12 sultanas we multiply by $3$3. But we know the ratio of sultanas and nuts is always $4:7$4:7, which means there must be $7\times3=21$7×3=21 nuts in the pack.

**Reflect:** We started with the ratio $32:56$32:56, then found the simplified ratio $4:7$4:7, and used this to find the equivalent ratio $12:21$12:21 that was relevant to our problem. Notice that the three ratios are all equivalent, but only $4:7$4:7 is a simplified ratio.

The ratio of milk to flour in a recipe is $3:4$3:4. If we use $5$5 cups of milk, how much flour is required?

**Think:** We have a ratio of $3:4$3:4 and want an equivalent ratio of $5:\editable{}$5:. However, $5$5 is not a nice multiple of $3$3. We can approach this problem in two ways:

**Method 1.** Solve step by step as we did for exchange rates, using a unitary method as follows:

**Do:**

$3$3 | $:$: | $4$4 |
Write the given ratio. |

$1$1 | $:$: | $\frac{4}{3}$43 |
Divide by $3$3 to find the amount of flour per $1$1 cup of milk. |

$1\times5$1×5 | $:$: | $\frac{4}{3}\times5$43×5 |
Multiply by $5$5 to find the amount of flour for $5$5 cups of milk. |

$5$5 | $:$: | $\frac{20}{3}$203 |

Thus, we need $\frac{20}{3}=6\frac{2}{3}$203=623 cups of flour.

**Method 2.** Use our understanding of ratios to convert the ratio to a fraction

**Think:** The ratio of milk to flour as $3:4$3:4 tells us:

- the amount of milk is $\frac{3}{4}$34 times the amount of flour
- the amount of flour is $\frac{4}{3}$43 times the amount of milk

Since we know the amount of milk we can simply multiply by the fraction $\frac{4}{3}$43 to find the amount of flour.

**Do:**

Amount of flour | $=$= | $5\times\frac{4}{3}$5×43 |

$=$= | $\frac{20}{3}$203 |

So again we get the result of $\frac{20}{3}$203 cups of flour but in less steps.

The two quantities are in proportion. Find the missing value.

- $\frac{\editable{}}{10}:\frac{35}{50}$10:3550

The ratio of bikes to helmets is $7:4$7:4. If there are $21$21 bikes, how many helmets are there?

To make the perfect shade of green for her painting, Xanthe knows she needs to mix blue to yellow in the ratio $\frac{4}{7}:\frac{5}{8}$47:58.

How much yellow paint will she need if she wants to use $4$4 pots of blue paint?

If the ratio of milk to flour in a recipe is $5:3$5:3. If we use $6$6 cups of milk, how much flour is required?

$5$5 : $3$3 $\div$÷$\editable{}$ $\div$÷$5$5 $1$1 **:**$\editable{}$ $\times$×$\editable{}$ $\times$×$\editable{}$ $6$6 : $\editable{}$

Concrete for a foundation is to be made from a mixture of sand, cement and aggregate in the ratio of $1:2:7$1:2:7.

A batch uses $3$3 kg of cement.

Calculate how much sand will be needed per batch.

Calculate how much aggregate will be needed per batch.

Calculate how much concrete will be made in total per batch.

find the ratio of two quantities