Ratio And Proportion
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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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연습 문제
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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