- Write and simplify a ratio
- Find an unknown term of a proportion by cross-multiplication
- Tell direct from inverse proportion and solve problems with them
- Divide a quantity in a given ratio and solve scale and percentage problems
Grandma uses 600 g of rice to cook plov for 4 people. How much rice does she need for 10 guests? How many kilometres long is a road that measures 4.2 cm on a map? What does a shirt cost after a 15% discount? All these questions are solved with the same tool: ratios and proportions. In this lesson you will learn to solve such problems quickly and without mistakes.
What is a ratio?
The quotient of two numbers, written a : b or a/b (b ≠ 0). A ratio shows how many times larger the first number is than the second, or what part of the second it makes up. a and b are called the terms of the ratio.
- a, bthe terms of the ratio
- kany non-zero number
The basic property of a ratio: both terms may be multiplied or divided by the same number. That is why a ratio can be simplified just like a fraction.
A class has 12 boys and 18 girls. The ratio of boys to girls is 12 : 18 = 2 : 3 (both terms were divided by 6): for every 2 boys there are 3 girls. The boys make up 12/30 = 2/5 of the whole class.
Proportions and cross-multiplication
A statement that two ratios are equal: a : b = c : d. a and d are the extremes of the proportion, b and c are its means. For example, 2 : 3 = 8 : 12.
- a, dthe extremes
- b, cthe means
The main property of a proportion: the product of the extremes equals the product of the means. It lets you find any unknown term, for example d = b · c / a.
Solve:
a) x : 12 = 5 : 3
b) (x + 1)/4 = (2x − 3)/3
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Check: 20 : 12 = 5 : 3, because both ratios equal 5/3.
b) Cross-multiply: 3(x + 1) = 4(2x − 3).
3x + 3 = 8x − 12 ⇒ 15 = 5x ⇒ x = 3.
Check: (3 + 1)/4 = 1 and (2 · 3 − 3)/3 = 1.
Direct and inverse proportion
Two quantities are directly proportional if multiplying one by some number multiplies the other by the same number: the amount of goods and the price paid, or distance and time at a constant speed. They are inversely proportional if multiplying one by a number divides the other by that number: speed and time over a fixed distance, or the number of workers and the time needed for the same job.
- x, ydirectly proportional quantities
- kthe constant of proportionality (k ≠ 0): the ratio y/x always equals k
The graph of a direct proportion is a straight line through the origin.
- x, yinversely proportional quantities
- kthe constant of proportionality: the product x · y always equals k
The graph of an inverse proportion is a hyperbola.
| Property | Direct proportion | Inverse proportion |
|---|---|---|
| Formula | y = kx | y = k/x |
| x is doubled | y is doubled | y is halved |
| What stays constant | y : x = k | x · y = k |
| Graph | a straight line through the origin | a hyperbola |
| Example | amount of goods and its cost | number of workers and time for a job |
Plov for 4 people needs 600 g of rice. How much rice is needed for 10 people?
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4 people — 600 g
10 people — x g
4/10 = 600/x ⇒ x = 600 · 10 / 4 = 1500 g = 1.5 kg.
6 workers build a fence in 12 days. How many days would 8 workers need, working at the same rate?
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6 · 12 = 8 · x ⇒ x = 72 / 8 = 9 days.
Common-sense check: there are more workers, so the number of days had to go down, and it did, from 12 to 9.
Dividing in a ratio, scale and percentages
- Sthe quantity being shared (the whole)
- a, b, cthe terms of the ratio a : b : c
- xthe share that corresponds to a
First find one part, S / (a + b + c), then multiply it by each term of the ratio.
Murad and Elvin earned 120 manat together and want to share it in the ratio 3 : 5. How much does each of them get?
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One part: 120 ÷ 8 = 15 manat.
Murad: 3 · 15 = 45 manat, Elvin: 5 · 15 = 75 manat.
Check: 45 + 75 = 120 and 45 : 75 = 3 : 5.
- nthe scale factor: 1 cm on the map is n cm in reality
Example: a road of 4.2 cm on a 1 : 500,000 map is 4.2 · 500,000 = 2,100,000 cm = 21 km long.
- athe part
- bthe whole (100%)
- pthe percentage
All three basic percentage problems are solved with this one proportion: whichever of the three values is unknown, find it by cross-multiplication.
a) Shoes costing 80 manat are reduced by 15%. What is the new price?
b) After a 20% increase the price is 54 manat. What was the original price?
c) 18 out of 24 students passed the exam. What percentage is that?
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b) 54 manat is 120% of the original price: x · 1.2 = 54 ⇒ x = 54 ÷ 1.2 = 45 manat.
c) 18/24 = p/100 ⇒ p = 18 · 100 / 24 = 75%.
Key points
- A ratio is the quotient of two numbers; both terms may be multiplied or divided by the same number: 12 : 18 = 2 : 3.
- In a proportion the product of the extremes equals the product of the means: a : b = c : d ⇔ ad = bc.
- In a direct proportion y = kx and the ratio is constant; in an inverse proportion y = k/x and the product is constant.
- To divide S in the ratio a : b, find one part S/(a + b) and multiply it by each term.
- A p% increase is · (1 + p/100), a p% decrease is · (1 − p/100); for successive changes the multipliers are multiplied.
Check yourself
10 questions. Every correct answer earns XP.