Working With Ratios

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Notes

Understanding Ratios

A **ratio** compares one part of a whole to another part (e.g., 2 : 5).
The order matters: the first quantity mentioned corresponds to the first number in the ratio.
The total number of parts is the sum of all numbers in the ratio (e.g., 4 : 3 gives 7 parts).
Ratios are different from fractions: a ratio compares parts to parts, a fraction compares a part to the whole.

Equivalent Ratios

**Equivalent ratios** represent the same proportion (e.g., 5 : 10 is equivalent to 20 : 40).
Multiply or divide each part of the ratio by the same number to find an equivalent ratio.
Scaling up (multiplying) gives larger numbers that may be more realistic in context.
Scaling down (dividing) leads to simplification.

Simplifying Ratios

A ratio is in **simplest form** when all numbers are integers with no common factor greater than 1.
Divide each part by the **highest common factor (HCF)** to simplify in one step.
If the HCF is not used, repeat the division process until the ratio is fully simplified.
Example: 30 : 18 simplifies to 5 : $3 (HCF = 6)$ .

Sharing an Amount in a Given Ratio

Add the parts to find the **total number of parts**.
Divide the total amount by the total number of parts to find the **value of one part**.
Multiply the value of one part by each part of the ratio to find each share.
Check that the shares add up to the original total.

Problem Solving: Difference Given

Find the difference in the number of parts between the two quantities.
Equate this difference to the given actual difference to find the value of one part.
Then multiply to find each quantity or the total.
Example: If ratio is 7 : 3 and Alfred eats 12 more than Bob, then 4 parts $= 12$ , so 1 part $= 3$ .

Problem Solving: One Quantity Given

Compare the given quantity to its corresponding number of parts in the ratio.
Divide to find the value of one part.
Multiply by the other part(s) to find the unknown quantity(ies).
Example: If red : white $= 3$ : 2 and Mark has 36 L red, then 3 parts $= 36 L$ , so 1 part $= 12 L$ , white $= 24 L$ .

Combining Two Ratios into a Three-Part Ratio

Identify the common quantity that appears in both ratios (the link).
Find equivalent ratios so that the link has the same value in both.
Then write the three-part ratio combining the two.
Example: $B:S = 5:2$ and $S:W = 6:7$ → make $S = 6$ in both: $B:S = 15:6, S:W = 6:7$ → $B:S:W = 15:6:7$ .

Direct Proportion

**Direct proportion**: as one quantity increases, the other increases by the same factor.
The ratio between the two quantities remains constant.
To solve, find the factor $(new \div old)$ and multiply the other quantity by that factor.
The **unitary method** (find 1 unit first) can also be used.

Inverse Proportion

**Inverse proportion**: as one quantity increases, the other decreases by the same factor.
To solve, find the factor $(new \div old)$ and divide the other quantity by that factor.
The unitary method: find the value for 1 unit (opposite operation), then scale.
Example: 3 pumps take 12 hours; 9 pumps take $12 \div (\frac{9}{3}) = 4$ hours.

Best Value Problems

Compare prices per **unit** (e.g., per kg, per litre) to find the best value.
Divide the cost by the quantity to get the unit price.
The lowest unit price is the best value for money.
Sometimes rounding up is necessary (e.g., number of tins of paint).

Ratio as Parts of a Whole

Difference in Ratio Problems

Combining Two Ratios

Direct vs Inverse Proportion

Practice questions

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1.A plane has 14 First Class seats, 70 Premium seats and 168 Economy seats. Find the ratio First Class seats : Premium seats : Economy seats in its simplest form.
Easy
- A1 : 5 : 12
- B2 : 10 : 24
- C14 : 70 : 168
- D7 : 35 : 84
2.Alex and Chris share sweets in the ratio Alex : Chris $= 7$ : 3. Alex receives 20 more sweets than Chris. Work out the number of sweets Chris receives.
Easy
- A15
- B20
- C12
- D18
3.Divide $24 in the ratio 7 : 5.
Easy
- A$14 and $10
- B$12 and $12
- C$7 and $5
- D$16.80 and $7.20
4.Kristian and Stephanie share some money in the ratio 3 : 2. Kristian receives $72. Work out how much Stephanie receives.
Easy
- A$48
- B$36
- C$24
- D$60
5.One day, the newspaper had 60 pages of news and advertisements. The ratio number of pages of news : number of pages of advertisements $= 5:7$ . Calculate the number of pages of advertisements.
Medium
- A35
- B25
- C42
- D30
6.Marianne sells photos. The selling price of each photo is $6. The selling price for each photo is made up of two parts, printing cost and profit. For each photo, the ratio printing cost : profit $= 5$ : 3. Calculate the profit she makes on each photo.
Medium
- A$2.25
- B$3.75
- C$2.50
- D$3.00
7.The Muller family are on holiday in New Zealand. The family visit two waterfalls, the Humboldt Falls and the Bridal Veil Falls. The ratio of the heights Humboldt Falls : Bridal Veil Falls $= 5$ : 1. The Humboldt Falls are 220 m higher than the Bridal Veil Falls. Calculate the height of the Humboldt Falls.
Medium
- A275 m
- B220 m
- C330 m
- D440 m
8.Adele, Barbara and Collette share $680 in the ratio 9 : 7 : 4. Show that Adele receives $306. Calculate the amount that Barbara and Collette each receives.
Medium
- ABarbara $238, Collette $136
- BBarbara $272, Collette $102
- CBarbara $224, Collette $150
- DBarbara $252, Collette $122

Summary

## Ratios - A **ratio** compares one part to another part (e.g., 2:5). - The order matters: the first item mentioned corresponds to the first number. - **Equivalent ratios** are found by multiplying/dividing all parts by the same number. - **Simplifying** a ratio: divide all parts by their highest common factor (HCF). ### Sharing an amount in a given ratio 1. Add the parts to find total parts. 2. Divide the total amount by total parts → value of 1 part. 3. Multiply each part by that value. ### Difference given - Find the difference in parts; equate to the actual difference → find 1 part. ### One quantity given - Compare given quantity to its parts → find 1 part; then find the other. ### Combining two ratios (three-part ratio) - Make the common quantity (e.g., women) the same in both ratios by scaling. - Then combine into a single three-part ratio. ## Proportion ### Direct proportion - Both quantities increase/decrease by the same factor. - Use unitary method: find 1 unit, then multiply. - Or find factor = new/old, then multiply the other quantity. ### Inverse proportion - As one quantity increases, the other decreases by the same factor. - Use unitary method: find 1 unit (multiply), then divide for the required quantity. - Or find factor = new/old, then divide the other quantity. ### Best value - Find the price per unit (e.g., per kg) for each option; the lowest is best. ## Key Formulas - **Ratio to fraction**: part/whole (e.g., in ratio 5:3, first part = 5/8 of whole). - **Total parts** = sum of ratio numbers. - **Value of 1 part** = total amount ÷ total parts. ## Tips - Label your ratios (e.g., A:B = 3:4). - For inverse proportion, think: more workers → less time. - Round up when buying items (e.g., tins of paint).

Key terms

Ratio: A way of comparing one part of a whole to another, written with colons (e.g., 2:5).
Equivalent ratio: Two ratios that represent the same proportion, found by multiplying or dividing all parts by the same number.
Simplest form: A ratio where all values are integers with no common factors (except 1).
Direct proportion: As one quantity increases/decreases, the other does so by the same factor; the ratio is constant.
Inverse proportion: As one quantity increases, the other decreases by the same factor (and vice versa).
Unitary method: Finding the value of one unit (e.g., 1 part) then scaling up/down to find the required amount.

Slides

What is a Ratio?

A ratio compares one part of a whole to another part.
Ratios are written with colons, e.g. 2 : 5 or 4 : 2 : 3.
The order matters: the first quantity mentioned matches the first number in the ratio.

In a recipe, flour and butter are in the ratio 2 : 1. Which statement is correct?

Flour is 2 parts, butter is 1 part.
Flour is 1 part, butter is 2 parts.
The whole is 2 parts.
Flour and butter are equal.

Equivalent Ratios

Equivalent ratios represent the same proportion.
Multiply or divide each part by the same number to get an equivalent ratio.
Example: 3 : 4 is equivalent to 15 : 20 (multiply by 5).

To find an equivalent ratio to 2 : 3, multiply both parts by 4 to $get ____$ : ____.

Simplifying Ratios

A ratio is in simplest form when all numbers are integers with no common factors.
Divide each part by the highest common factor (HCF).
Example: 30 : 18 simplifies to 5 : 3 (divide by 6).

What is the simplest form of the ratio 45 : 30?

9 : 6
3 : 2
15 : 10
5 : 3

Sharing an Amount in a Given Ratio

Add the parts to find the total number of parts.
Divide the total amount by the total parts to find the value of one part.
Multiply each part of the ratio by the value of one part.

Share $24 in the ratio 7 : 5. One part = $24 $\div (7+5) =$ $____. The larger share is $____.

Difference Between Two Parts

If one quantity is given as a difference, find the difference in parts.
Divide the actual difference by the difference in parts to find the value of one part.
Then multiply to find each quantity.

Alex and Chris share sweets in the ratio 7 : 3. Alex gets 20 more sweets than Chris. How many sweets does Chris get?

Given One Quantity, Find the Other

Compare the given quantity to its number of parts.
Find the value of one part by division.
Multiply by the other part's number to find the unknown quantity.

Kristian and Stephanie share money in the ratio 3 : 2. Kristian gets $72. One part = $72 $\div 3 =$ $____. Stephanie gets $____.

Combining Two Ratios into a Three-Part Ratio

Find the common quantity that appears in both ratios.
Make the value of that common quantity the same in both ratios.
Then write the three-part ratio.

Which of these are true about combining ratios?

If $A:B = 5:2$ and $B:C = 6:7$ , then $A:B:C = 15:6:7$ .
The common quantity must be made the same in both ratios.
You can only combine two-part ratios into a three-part ratio.
The combined ratio is always in simplest form.

Direct Proportion

As one quantity increases, the other increases by the same factor.
The ratio between the two quantities is constant.
Use the unitary method: find one unit, then multiply.

If 2 boxes of cereal weigh 800 g, how much do 5 boxes weigh?

1600 g
2000 g
2400 g
3200 g

Inverse Proportion

As one quantity increases, the other decreases by the same factor.
Example: more workers means less time to complete a job.
Use the unitary method: find the value for one unit, then divide.

If 3 pumps fill a pool in 12 hours, 1 pump takes ____ hours. So 9 pumps take ____ hours.

Best Value for Money

Compare the price per unit (e.g., per kg) for different options.
Divide the total price by the quantity to find the unit price.
The lowest unit price is the best value.

Which is better value: 1.5 kg flour for $0.84 or 5 kg flour for $2.65?

1.5 kg for $0.84
5 kg for $2.65
They are the same value
Cannot tell

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