Ratio And Proportion

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Notes

Introduction to Ratios

A **ratio** compares one part of a whole to another part, using a colon (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 fraction compares a part to the whole, while a ratio compares parts to each other.
Example: A pizza shared 5 slices to 3 slices gives ratio 5 : 3; fractions are $\frac{5}{8}$ and 3/8.

Equivalent & Simplified Ratios

**Equivalent ratios** represent the same proportion; multiply or divide all parts by the same number.
Example: 3 : 2 is equivalent to 300 : 200 (multiply by 100).
To find an equivalent ratio when one part is known, divide the known value by its ratio part to find the multiplier.
**Simplifying** a ratio means dividing all parts by a common factor (preferably the HCF).
Example: 30 : 18 simplifies to 5 : 3 (divide by 6).
A ratio is in simplest form when all numbers are integers with no common factor greater than 1.

Sharing in a Ratio

To share an amount in a given ratio: add the parts to find total parts, divide the amount by total parts to find one part's value.
Multiply one part's value by each ratio number to find each share.
Always check that the shares sum to the original amount.
Example: Share $200 in ratio 5 : 3 → $5+3=8$ parts, $200÷8=$25 per part, so $125 and $75.

Problem Solving with Ratios

When given the **difference** between two quantities, find the difference in parts, then equate to the actual difference to find one part.
Example: Alfred eats 12 more than Bob in ratio 7:3 → 4 parts $= 12$ , so 1 part = 3; Alfred gets 21, Bob gets 9.
When given **one quantity**, divide that quantity by its ratio number to find one part, then multiply for the other quantity.
To **combine two two-part ratios** into a three-part ratio, make the common quantity the same in both ratios.
Example: $B:S = 5:2$ and $S:W = 6:7$ → multiply first by 3 to get $B:S = 15:6$ , then combine to $B:S:W = 15:6:7$ .
Use ratios to find percentages: e.g., 15 out of 28 parts black ≈ 53.6%.

Direct Proportion

**Direct proportion**: as one quantity increases, the other increases by the same factor; the ratio is constant.
Solve by finding the factor $(new \div old)$ and multiplying the other quantity by that factor.
**Unitary method**: find the value for 1 unit, then multiply to the required number of units.
Example: 8 boxes weigh 60 kg → 1 $box = 7.5$ kg, so 7 boxes $= 52.5$ kg.
To find **best value**, calculate price per unit (e.g., per kg) and choose the lower unit price.

Inverse Proportion

**Inverse proportion**: as one quantity increases, the other decreases by the same factor.
Solve by finding the factor $(new \div old)$ and dividing the other quantity by that factor.
Unitary method: find the value for 1 unit (e.g., time for 1 worker) then scale by dividing.
Example: 3 pumps fill in 12 hours → 1 pump takes 36 hours, so 9 pumps take $36\div 9=4$ hours.
Think about context: more workers → less time (inverse); more boxes → more weight (direct).

Ratio as parts of a whole

Sharing in a ratio

Direct proportion example

Inverse proportion example

Practice questions

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1.When full, a cruise ship carries 880 guests and 360 crew. Write the ratio guests : crew in its simplest form.
Easy
- A22 : 9
- B88 : 36
- C44 : 18
- D11 : 9
2.A box contains 22 coloured pencils. 6 pencils are pink, 9 pencils are blue and 7 pencils are yellow. Write down the ratio pink pencils : not pink pencils in its simplest form.
Easy
- A6 : 16
- B3 : 8
- C6 : 22
- D3 : 11
3.Maia shares $3000 between her three children. She gives the eldest child $1200, the second eldest child $1000 and the rest to the youngest child. Write this information as a ratio in its simplest form (eldest : second : youngest).
Easy
- A12 : 10 : 8
- B6 : 5 : 4
- C1200 : 1000 : 800
- D3 : 2 : 1
4.Divide 120 in the ratio 1 : 2. What are the two numbers?
Easy
- A40 and 80
- B30 and 90
- C60 and 60
- D20 and 100
5.Alan and Beth share $1190 in the ratio Alan : Beth $= 5$ : 2. How much does Alan receive?
Easy
- A$850
- B$340
- C$595
- D$700
6.Alex and Chris share sweets in the ratio Alex : Chris $= 7$ : 3. Alex receives 20 more sweets than Chris. How many sweets does Chris receive?
Medium
- A15
- B35
- C20
- D25
7.A car park has 880 parking spaces. The ratio of reserved spaces : not reserved spaces $= 1$ : 10. How many spaces are not reserved?
Medium
- A80
- B800
- C880
8.Jess and Adam share $420 in the ratio 5 : 1. How much does Jess receive?
Medium
- A$350
- B$70
- C$300
- D$210

Summary

## Ratio & Proportion - Quick Revision ### Introduction to Ratios - A ratio compares one part to another part (e.g., 2:5, 3:1, 4:2:3). - The order matters: the first item mentioned corresponds to the first number. - Total parts = sum of all parts in the ratio. ### Equivalent & Simplified Ratios - **Equivalent ratios** are found by multiplying or dividing all parts by the same number. - **Simplified ratio**: all values are integers with no common factors (use HCF). ### Sharing in a Ratio 1. Add the parts to find total parts. 2. Divide the total amount by total parts to find value of 1 part. 3. Multiply each part by the value of 1 part. ### Problem Solving with Ratios - **Given difference**: find difference in parts, equate to given difference, find 1 part, then compute. - **Given one quantity**: compare given quantity to its parts, find 1 part, then find others. - **Combining ratios**: make the common term equal in both ratios, then combine into a three-part ratio. ### Direct Proportion - As one quantity increases, the other increases by the same factor. - **Unitary method**: find value of 1 unit, then multiply. - **Best value**: compare price per unit (e.g., per kg). ### Inverse Proportion - As one quantity increases, the other decreases by the same factor. - **Unitary method**: find time for 1 worker (multiply), then divide by new number. ### Key Formulas - **Direct proportion**: if y ∝ x, then y = kx (k constant). - **Inverse proportion**: if y ∝ 1/x, then y = k/x (k constant).

Key terms

Ratio: A way of comparing one part of a whole to another, written using a colon (e.g., 2:5).
Equivalent ratio: Ratios that represent the same proportion, obtained by multiplying or dividing all parts by the same number.
Simplified ratio: A ratio in its simplest form where all values are integers with no common factors (other than 1).
Direct proportion: Two quantities increase or decrease at the same rate; the ratio between them is constant.
Inverse proportion: As one quantity increases, the other decreases at the same rate; the product of the two quantities is constant.
Unitary method: Finding the value of one unit first, then scaling up or down to find the required value.

Slides

Introduction to Ratios

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

In a ratio 3 : 7, how many parts are there in total?

Equivalent & Simplified Ratios

Equivalent ratios represent the same proportion; multiply or divide all parts by the same number.
To simplify a ratio, divide all parts by a common factor (preferably the HCF).
Example: 30 : 18 simplifies to 5 : 3 by dividing by 6.

Simplify the ratio 45 : 30. The simplest form $is ____$ : ____.

Sharing in a Ratio

Add the parts to find total number of parts.
Divide the total amount by the 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.

Share $120 in the ratio 1 : 2. How much does the first person get?

Problem Solving with Ratios (Difference Given)

If the difference between two quantities is given, find the difference in parts.
Divide the actual difference by the part difference to find one part.
Then multiply to find each quantity.

Alan and Beth share money in ratio 5:2. Alan gets $30 more than Beth. How much does Beth get? Beth gets $____.

Problem Solving with Ratios (One Quantity Given)

If one quantity is known, compare it to its number of parts to find the value of one part.
Then multiply to find the other quantity or total.

The ratio of flour to sugar is 3:2. If 24 kg of flour is used, how much sugar is needed?

16 kg
12 kg
8 kg
36 kg

Combining Ratios

To combine two two-part ratios into a three-part ratio, make the common part the same.
Find equivalent ratios so the linking quantity has the same number of parts.
Then write the three-part ratio.

Which of these are true about combining ratios?

You multiply the linking part to make it equal in both ratios.
The final ratio has three numbers.
You add the two ratios together.

Direct Proportion

Direct proportion: as one quantity increases, the other increases by the same factor.
The ratio between the quantities is constant.
Use the unitary method: find the value of one unit, then multiply.

If 8 boxes weigh 60 kg, how much do 7 boxes weigh? $____ kg$ .

Inverse Proportion

Inverse proportion: as one quantity increases, the other decreases by the same factor.
The product of the two quantities is constant.
Use the unitary method: find the value for one unit, then divide.

If 3 pumps fill a pool in 12 hours, how long would 9 pumps take?

4 hours
36 hours
6 hours
3 hours

Best Value Problems

To find best value, calculate the price per unit (e.g. per kg) for each option.
The option with the lowest unit price is the best value.
Always compare the same unit size.

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

1.5 kg for $0.84
5 kg for $2.65
They are equal
Cannot tell

Flashcards

What is a ratio?

A ratio is a way of comparing one part of a whole to another, using numbers separated by a colon (e.g., 2:5).

How do you simplify a ratio?

Divide each part of the ratio by a common factor, ideally the highest common factor (HCF). For example, 30:18 simplifies to 5:3 by dividing by 6.

How do you share an amount in a given ratio?

Add the parts to find the total number of parts, divide the amount by that total to find the value of one part, then multiply each part of the ratio by that value.

Write the ratio 880 guests : 360 crew in simplest form.

Divide both by 40: 22:9.

A box has 6 pink, 9 blue, 7 yellow pencils. Write the ratio pink : not pink in simplest form.

Not pink $= 9+7=16$ , so pink:not pink $= 6:16 = 3:8$ .

Divide 120 in the ratio 1:2.

Total parts $= 3$ , one part $= 40$ , so 40:80.

Alan and Beth share $1190 in ratio 5:2. How much does Alan receive?

Total parts $= 7$ , one part = $170, Alan gets $5\times 170 =$ $850.

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

Difference in parts $= 4$ , so 4 parts $= 20$ , one part $= 5$ . Chris gets $3\times 5 = 15$ sweets.

Share $1027 in ratio 7:4:2. How much does each get?

Total parts $= 13$ , one part = $79. George: 7×79=$553, Louis: 4×79=$316, Beatriz: 2×79=$158.

Share $60000 in ratio 2:6:7. How much does each get?

Total parts $= 15$ , one part = $4000. Alexa: 2×4000=$8000, Ben: 6×4000=$24000, Chloe: 7×4000=$28000.

Share 518 in ratio 2:5.

Total parts $= 7$ , one part $= 74$ , so 148:370.

Asif and Ben share $2100 in ratio 3:7. How much does Asif get?

Total parts $= 10$ , one part = $210, Asif gets $3\times 210 =$ $630.

The ratio $flights:hotels = 3:8$ . Flights cost $861. Find total cost.

3 parts = $861, so one part = $287. Hotels $= 8\times 287 =$ $2296. Total $= 861+2296 =$ $3157.

A car park has 880 spaces. Ratio reserved:not reserved $= 1:10$ . How many not reserved?

Total parts $= 11$ , one part $= 80$ , not reserved $= 10\times 80 = 800$ .

Jess and Adam share $420 in ratio 5:1. How much each?

Total parts $= 6$ , one part = $70. Jess: 5×70=$350, Adam: 1×70=$70.

Ratio of seats $A:B:C = 152:171:57$ . Simplify.

Divide by 19: 8:9:3.

Ticket prices ratio $A:B:C = 9:7:4$ . Section C costs $6. Find cost of A and B.

4 parts = $6, one part = $1.50. B: 7×1.50=$10.50, A: 9×1.50=$13.50.

A bag has yellow, pink, green balls in ratio 7:3:5. There are 42 yellow balls. Find total balls.

7 parts $= 42$ , one part $= 6$ . Total parts $= 15$ , total balls $= 15\times 6 = 90$ .

100g cereal $= 360$ kcal, 100ml milk $= 45$ kcal. Sasha has 35g cereal and 180ml milk. Total kcal?

Cereal: $\frac{35}{100}\times 360 = 126$ kcal. Milk: $\frac{180}{100}\times 45 = 81$ kcal. Total $= 126+81 = 207$ kcal.

Ratio $red:green:blue = 12:5:2$ . There are 112 more red than green. Find number of blue.

Difference in parts $red-green = 7$ , so 7 parts $= 112$ , one part $= 16$ . Blue $= 2\times 16 = 32$ .

45 horses and 20 cows. Write ratio horses:cows in simplest form.

Divide by 5: 9:4.

Farmer has 45 horses and wants ratio $horses:cows = 5:3$ . He keeps horses and buys more cows. How many cows must he buy?

45 horses $= 5$ parts, so one part $= 9$ . Cows needed $= 3\times 9 = 27$ . He has 20, so buy 7 more.

T-shirts ratio $white:blue:green = 5:4:1$ . 48 are blue. Find total T-shirts.

4 parts $= 48$ , one part $= 12$ . Total parts $= 10$ , total $= 10\times 12 = 120$ .

Raoul sells watch for $36. Ratio price $paid:profit = 7:2$ . Find profit.

Total parts $= 9$ , one part = $4, profit $= 2\times 4 =$ $8.

Geri and Tom pay start-up costs in ratio 23:11. Tom pays $135000 less than Geri. How much each?

Difference in parts $= 12$ , so 12 parts = $135000, one part = $11250. Geri: 23×11250=$258750, Tom: 11×11250=$123750.

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