Ratio And Proportion

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Notas de aula

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 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 ÷ 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 ÷ 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÷9=4 hours.
Think about context: more workers → less time (inverse); more boxes → more weight (direct).

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Questões de prática

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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).
Medium
- 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?
Medium
- 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?
Easy
- A$350
- B$70
- C$300
- D$210

Mais perguntas

9.Zach is looking at the costs of a holiday. The ratio of costs flights : hotels = 3 : 8. The cost of the flights is $861. What is the total cost of flights and hotels?
Medium
- A$3157
- B$2296
- C$1148
10.A bag contains yellow, pink and green balls only. The ratio yellow : pink : green = 7 : 3 : 5. There are 42 yellow balls. How many balls in total?
Medium
- A90
- B60
- C72
- D84
11.A bag contains red, green and blue marbles only. The ratio red : green : blue = 12 : 5 : 2. There are 112 more red marbles than green marbles. How many blue marbles are there?
Hard
- A32
- B16
- C28
- D40
12.A farmer has 45 horses and 20 cows. He wants the ratio of horses : cows to equal 5 : 3. He keeps his 45 horses and buys some more cows. How many cows must he buy?
Medium
- A7
- B5
- C10
- D3
13.Nadira has T-shirts that are either white, blue or green. The ratio white : blue : green = 5 : 4 : 1. 48 of the T-shirts are blue. How many T-shirts in total?
Medium
- A120
- B96
- C60
- D144
14.Raoul sells his watch for $36. The ratio price paid : profit = 7 : 2. How much profit does he make?
Medium
- A$8
- B$10
- C$12
- D$6
15.A recipe for 12 pancakes uses 110 g flour, 2 eggs, 200 ml milk and 50 g butter. How much flour is needed for 30 pancakes?
Medium
- A275 g
- B220 g
- C330 g
- D165 g
16.Asif and Ben share $2100 in the ratio Asif : Ben = 3 : 7. How much does Asif receive?
Easy
- A$630
- B$900
- C$1470
- D$700
17.A map has a scale of 1 : 50,000. Two towns are 8 cm apart on the map. What is the actual distance in km?
Medium
- A4 km
- B40 km
- C0.4 km
- D400 km
18.Three friends share a prize in the ratio 2 : 3 : 5. The middle friend receives $120. How much does the largest share receive?
Medium
- A$200
- B$240
- C$300
- D$360
19.A factory produces 2400 bolts in 8 hours at a constant rate. How many bolts are produced in 5 hours?
Easy
- A1500
- B1200
- C1800
- D2000
20.It takes 6 workers 10 days to build a wall. How many days would it take 15 workers to build the same wall?
Medium
- A4
- B5
- C6
- D3
21.A rectangle has length 12 cm and width 8 cm. Find the ratio of its length to its perimeter.
Medium
- A12 : 40
- B12 : 20
- C12 : 96
- D12 : 24
22.What is a ratio?
Easy
- AA way of comparing one part of a whole to another
- BA way of comparing a part to the whole
- CA way of comparing two whole numbers
- DA way of comparing a quantity to a unit
23.Which of the following is an equivalent ratio to 2 : 5?
Easy
- A4 : 10
- B3 : 6
- C1 : 3
- D5 : 2
24.Simplify the ratio 12 : 8.
Easy
- A3 : 2
- B6 : 4
- C4 : 3
- D2 : 1
25.A bag contains 5 red marbles and 3 blue marbles. What is the ratio of red marbles to blue marbles?
Easy
- A5 : 3
- B3 : 5
- C5 : 8
- D3 : 8
26.Share $80 in the ratio 3 : 5. What is the larger share?
Medium
- A$50
- B$30
- C$40
- D$48
27.The ratio of boys to girls in a class is 4 : 5. There are 20 boys. How many students are in the class?
Medium
- A45
- B25
- C36
- D40
28.A map has a scale of 1 : 25000. Two towns are 5 cm apart on the map. What is the actual distance in km?
Medium
- A1.25 km
- B12.5 km
- C125 km
- D0.125 km
29.It takes 4 workers 6 days to paint a house. How long would it take 8 workers to paint the same house?
Medium
- A3 days
- B12 days
- C2 days
- D4 days
30.A recipe for 6 pancakes uses 150 g flour. How much flour is needed for 10 pancakes?
Medium
- A250 g
- B200 g
- C300 g
- D180 g
31.The ratio of A : B is 3 : 4 and the ratio of B : C is 2 : 5. Find the ratio A : B : C.
Hard
- A3 : 4 : 10
- B6 : 8 : 20
- C3 : 2 : 5
- D6 : 4 : 5
32.Geri and Tom share start-up costs in the ratio 23 : 11. Tom pays $135,000 less than Geri. How much does Geri pay?
Hard
- A$258,750
- B$123,750
- C$135,000
- D$202,500
33.If 3 kg of apples cost $6, how much do 5 kg cost?
Easy
- A$10
- B$9
- C$12
- D$8
34.Which of the following is the simplest form of the ratio 18 : 24?
Easy
- A3 : 4
- B6 : 8
- C9 : 12
- D2 : 3
35.A paint mixture uses red and white paint in the ratio 3 : 2. How much red paint is needed to make 20 litres of the mixture?
Medium
- A12 litres
- B8 litres
- C15 litres
- D10 litres
36.Which of the following is a ratio?
Easy
- A3 : 4
- B3 + 4
- C3 × 4
- D3 = 4
37.A ratio compares a part to the whole.
Easy
True or false?
38.Simplify the ratio 24 : 36.
Easy
- A2 : 3
- B4 : 6
- C12 : 18
- D6 : 9
39.If 5 kg of potatoes cost $8, how much do 15 kg cost?
Easy
- A$24
- B$16
- C$40
- D$8
40.If 4 workers can paint a fence in 6 hours, then 8 workers can paint the same fence in 3 hours, assuming they work at the same rate.
Medium
True or false?
41.It takes 6 hours to fill a tank using 3 pumps. How long would it take to fill the same tank using 5 pumps?
Medium
- A3.6 hours
- B10 hours
- C2.5 hours
- D7.5 hours
42.Match each ratio to its simplified form.
Easy
- 8 : 12
- 15 : 20
- 18 : 24
- 2 : 3
- 3 : 4
- 3 : 4
43.Arrange these steps in order to solve: 'Divide $240 in the ratio 3 : 5'.
Medium
- Find total parts: 3+5=8
- Find one part: $240÷8=$30
- Smaller share: 3×$30=$90
- Larger share: 5×$30=$150
44.Arrange the following steps in the correct order to share $240 in the ratio 3:5.
Medium
- Add the parts: 3+5=8
- Divide total by total parts: 240÷8=30
- Multiply each part by 30: 3×30=90, 5×30=150
- Check: 90+150=240
45.Which of the following are equivalent to the ratio 4:6? (Select all that apply)
Medium
- A2:3
- B8:12
- C12:18
- D20:30
- E1:2
46.Fill in the blank to complete the sentence.
Easy
The ratio 2 : 3 can be written as the fraction ____.
47.Share $150 in the ratio 2 : 3. What is the value of the larger share?
Medium
48.The ratio of boys to girls in a school is 5 : 3. There are 240 more boys than girls. How many students are there in total?
Medium
49.Complete the sentence.
Medium
If y is directly proportional to x, and y = 10 when x = 2, then y = ____ when x = 5.
50.The ratio of adults to children on a bus is 5:3. There are 18 more adults than children. How many people are on the bus in total?
Medium
51.Explain what it means for two quantities to be in direct proportion.
Easy
52.Share $150 in the ratio 2 : 3. What is the value of the larger share?
Medium
- A$99
- B$108
- C$90
- D$81
53.The ratio of boys to girls in a school is 5 : 3. There are 240 more boys than girls. How many students are there in total?
Medium
- A864
- B1056
- C960
- D1152
54.The ratio of adults to children on a bus is 5:3. There are 18 more adults than children. How many people are on the bus in total?
Medium
- A72
- B86
- C65
- D79

Resumos de revisão

## 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).

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.

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 5×170 = $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 3×5 = 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 3×210 = $630.

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

3 parts = $861, so one part = $287. Hotels = 8×287 = $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 = 10×80 = 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 = 15×6 = 90.

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

Cereal: 35/100×360 = 126 kcal. Milk: 180/100×45 = 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 = 2×16 = 32.

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

Divide by 5: 9:4.

Farmer has 45 horses and 20 cows, and wants the ratio horses:cows = 5:3. He keeps all his horses and buys more cows. How many cows must he buy?

45 horses = 5 parts, so one part = 9. Cows needed = 3×9 = 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 = 10×12 = 120.

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

Total parts = 9, one part = $4, profit = 2×4 = $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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