A proportion is an equality of two ratios, for example, \(2:3 = 8:12\). To solve proportions, you only need one tool — the main property of a proportion: the product of the extreme terms equals the product of the mean terms. In school, it's called the "cross-multiplication" rule. Below, we explain what a ratio and a proportion are, how to find an unknown term, how direct proportionality differs from inverse, analyze problems about recipes, scales, and prices, cover percentages, and provide a trainer with 10 problems.
What is a ratio and a proportion
A ratio of two numbers is their quotient. The ratio \(2:3\) (read "two to three") shows how many times one number is greater than the other, or what part one is of the other. It can be written in two ways: \(2:3\) or as a fraction \(\frac{2}{3}\) — they mean the same thing.
A proportion is an equality of two ratios:
A proportion has four numbers, and they have names. The numbers at the ends of the expression are the extreme terms, and the numbers in the middle are the mean terms. A proportion is true when both ratios are equal to the same number.
The main property of a proportion: the "cross-multiplication" rule
The main property of a proportion: the product of the extreme terms equals the product of the mean terms.
Why does this happen? Multiply both sides of the equality \(\frac{a}{b} = \frac{c}{d}\) by \(b \cdot d\). The \(b\) will cancel out on the left, and the \(d\) on the right, leaving exactly \(a \cdot d = c \cdot b\). It's not magic — just a property of equalities.
This is precisely what schoolchildren call "solving proportions by cross-multiplication": numbers are multiplied diagonally, crosswise. The property also works in reverse: if for four numbers (none of them zero) \(a \cdot d = b \cdot c\) holds true, then a correct proportion can be formed from them.
This immediately gives us two skills: checking if a proportion is correct, and finding an unknown number in it.
How to find an unknown term of a proportion
From the main property, we derive two short rules:
- unknown extreme term = product of means : known extreme term;
- unknown mean term = product of extremes : known mean term.
Memorizing them by heart is not necessary. One universal method is sufficient: multiply crosswise, get an equation of the form "number \(\cdot\, x\) = number", and then divide. Let's break this down step-by-step using the proportion \(x : 7 = 12 : 21\).
The last step, checking, takes five seconds and catches almost any arithmetic error, so it's worth doing.
Direct and inverse proportionality
Two quantities are called directly proportional if, when one increases by a certain factor, the other increases by the same factor. Bought twice as many notebooks, paid twice as much. For such quantities, their ratio is constant.
Quantities are inversely proportional if, when one increases by a certain factor, the other decreases by the same factor. Twice as many workers means half the days for the same job. For such quantities, their product is constant.
The main difference is visible when we set up the proportion. For a direct relationship, it is set up "as is": \(\frac{a_1}{a_2} = \frac{b_1}{b_2}\). For an inverse relationship, the second ratio is inverted: \(\frac{a_1}{a_2} = \frac{b_2}{b_1}\).
Why? For an inverse relationship, the product is constant: \(a_1 \cdot b_1 = a_2 \cdot b_2\). Divide both sides by \(a_2 \cdot b_1\) — and you get exactly \(\frac{a_1}{a_2} = \frac{b_2}{b_1}\). So, the "inversion" is not a spell to remember, but a consequence of the product remaining constant, not the ratio.
| Characteristic | Direct | Inverse |
|---|---|---|
| Relationship | more → more | more → less |
| Constant | ratio a : b | product a × b |
| Setup | a₁ : a₂ = b₁ : b₂ | a₁ : a₂ = b₂ : b₁ |
| Example | 3 pens — 90 ₽, 5 pens — 150 ₽ | 4 painters — 9 days, 6 painters — 6 days |
How to set up a proportion for a problem: recipe, scale, price
Word problems involving proportions are solved using one scenario:
- Write down the conditions in two columns so that identical quantities are aligned vertically (portions under portions, grams under grams).
- Denote the unknown quantity with \(x\).
- Determine the type of relationship: mentally increase the first quantity and observe if the second one increases or decreases.
- Set up the proportion — for an inverse relationship, invert the second ratio.
- Solve using cross-multiplication and check if the answer is reasonable.
Let's illustrate with a recipe. For 4 servings of pancakes, you need 300 g of flour. How much flour is needed for 6 servings? More servings means more flour: the relationship is direct, so the proportion is set up "as is":
For 6 servings, 450 g of flour are needed. The answer is reasonable: the number of servings increased by half, and the amount of flour needed also increased by half.
Problems involving scale (the distance on a map relates to the distance on the ground in the same way as 1 to the scale number) and price (the quantity of goods relates to the quantity in the same way as the cost to the cost) are solved similarly. Full explanations are in the examples below.
Percentages using proportions
A percent is one-hundredth of a number, so any simple percentage problem can be turned into a proportion:
Somewhere, one of the four numbers is unknown — that's what we find using cross-multiplication.
Finding the part. What is 15% of 240? We set up \(x : 240 = 15 : 100\), which means \(100x = 240 \cdot 15 = 3600\), and \(x = 36\).
Finding the whole. The number 21 is 35% of some number. We set up \(21 : x = 35 : 100\), which means \(35x = 21 \cdot 100 = 2100\), and \(x = 60\).
Finding the percentage. What percentage is 18 of 45? We set up \(18 : 45 = p : 100\), which means \(45p = 18 \cdot 100 = 1800\), and \(p = 40\), meaning 40%.
One method — three different types of problems, nothing extra to memorize. The question "by what percentage did the quantity increase or decrease" is separate: first, find the change, and then divide it by the original value — this is already the formula for percentage change, not a simple proportion.
Examples of solving proportions with step-by-step explanations
Example 1. Unknown mean term
Problem. Find the unknown term of the proportion \(15 : x = 45 : 24\).
Step 1. Identify the terms. The extreme terms are 15 and 24; the mean terms are \(x\) and 45.
Step 2. Apply the main property: \(15 \cdot 24 = x \cdot 45\), which means \(360 = 45x\).
Step 3. Divide: \(x = 360 : 45 = 8\).
Check. \(15 : 8 = 1.875\) and \(45 : 24 = 1.875\) — the ratios are equal.
Answer: \(x = 8\).
Example 2. Direct proportionality: price and quantity
Problem. Six notebooks cost 210 ₽. How much do 10 identical notebooks cost?
Step 1. More notebooks means a higher cost, so the relationship is direct.
Step 2. Write down the conditions so that identical quantities are aligned, and set up the proportion "as is":
Step 3. Cross-multiplication: \(6x = 10 \cdot 210 = 2100\), so \(x = 2100 : 6 = 350\).
Check. One notebook costs \(210 : 6 = 35\) ₽, and ten cost \(35 \cdot 10 = 350\) ₽. It matches.
Answer: 350 ₽.
Example 3. Inverse proportionality: workers and days
Problem. Eight workers complete a job in 15 days. How many days will it take 12 workers to complete the same job if they work at the same pace?
Step 1. More workers means fewer days are needed. The relationship is inverse.
Step 2. Set up the proportion by inverting the second ratio:
Here, \(x\) represents the days for 12 workers, and 15 represents the days for 8 workers: they have swapped places because the relationship is inverse.
Step 3. Cross-multiplication: \(8 \cdot 15 = 12x\), which means \(120 = 12x\), and \(x = 10\).
Check. The product is constant: \(8 \cdot 15 = 120\) and \(12 \cdot 10 = 120\). It matches, and the answer is reasonable — fewer days are needed.
Answer: 10 days.
Example 4. Map scale
Problem. The map scale is \(1 : 200,000\). The distance between two villages on the map is 7 cm. What is the distance between them on the ground?
Step 1. A scale of \(1 : 200,000\) means: 1 cm on the map corresponds to 200,000 cm on the ground. This is a ready-made ratio.
Step 2. Set up the proportion:
Step 3. Cross-multiplication: \(1 \cdot x = 200,000 \cdot 7\), so \(x = 1,400,000\) cm.
Step 4. Convert to convenient units: one kilometer has 100,000 cm, so \(1,400,000 : 100,000 = 14\) km.
Answer: 14 km.
Example 5. Percentages using proportions
Problem. There are 25 students in a class, and 60% of them went on a field trip. How many students went on the trip?
Step 1. The whole is all 25 students, which corresponds to 100%. The unknown part is \(x\) students, corresponding to 60%.
Step 2. Set up the proportion "part : whole = percent : 100":
Step 3. Cross-multiplication: \(100x = 25 \cdot 60 = 1500\), so \(x = 15\).
Check. 15 students out of 25 is \(15 : 25 = 0.6\), which is exactly 60%.
Answer: 15 students.
Common mistakes when solving proportions
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Multiplying numbers "along" instead of crosswise: in the proportion 5 : 2 = 20 : 8, calculating 5 · 2 and 20 · 8.
The main property relates the extreme terms to the mean terms, i.e., numbers diagonally: 5 · 8 = 40 and 2 · 20 = 40. The products match — the proportion is correct. But 5 · 2 and 20 · 8 are not equal and tell us nothing.
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Setting up a "direct" proportion for an inverse relationship problem. Six pumps fill a pool in 10 hours; setting up 6 : 15 = 10 : x gives 25 hours for 15 pumps.
More pumps should mean less time, but the result is more time: this means the proportion was set up incorrectly. The second ratio should be inverted: 6 : 15 = x : 10, then 15x = 60, and x = 4 hours. Always check your answer for reasonableness.
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Dividing incorrectly after cross-multiplication: from the equation 8x = 96, getting x = 8 : 96.
To find x, divide the product by the coefficient next to x: x = 96 : 8 = 12. It's a good habit to substitute the found number back: 8 · 12 = 96, everything is correct.
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Not matching units of measurement. In a problem with a scale of 1 : 50,000 and 3 cm on the map, giving the answer "1.5" without units or calling it meters.
First, calculate in the units given by the scale: 3 · 50,000 = 150,000 cm. Only then convert: 150,000 cm = 1500 m = 1.5 km. Units in the answer are as important as the number itself.
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Confusing ratio and proportion: when asked to set up a proportion, writing a single ratio "12 : 4".
A ratio is a single quotient, e.g., 12 : 4. A proportion is an equality of two ratios; it must have four numbers and an equals sign: 12 : 4 = 30 : 10 (check: 12 · 10 = 120 and 4 · 30 = 120).
Questions and Answers
How does a ratio differ from a proportion?
A ratio is the quotient of two numbers, for example, 12 : 4. A proportion is an equality of two ratios; it has four numbers: 12 : 4 = 30 : 10. A ratio answers the question "how many times greater," while a proportion states that two such comparisons yield the same result.
What are the extreme and mean terms of a proportion?
In the expression a : b = c : d, the extreme terms are those at the ends, i.e., a and d. The mean terms are those in the middle, i.e., b and c. If the proportion is written as fractions, the extreme terms are the numerator of the first fraction and the denominator of the second, and the mean terms are the denominator of the first and the numerator of the second.
How to check if a proportion is correct?
Multiply crosswise and compare the products. For example, for 9 : 12 = 15 : 20, calculate 9 · 20 = 180 and 12 · 15 = 180 — the numbers match, so the proportion is correct. If the products are different, the proportion is incorrect.
Can the terms of a proportion be swapped?
Yes, if done according to the rules. In a correct proportion, you can swap the extreme terms, you can swap the mean terms, or you can invert both ratios — the proportion will remain correct. From 4 : 6 = 10 : 15, we get 4 : 10 = 6 : 15 or 6 : 4 = 15 : 10 — the product diagonally is 60 everywhere. However, you cannot swap just one number.
What does "solving a proportion by cross-multiplication" mean?
This is a colloquial term for the main property of a proportion. Numbers are multiplied diagonally, crosswise: extreme by extreme and mean by mean. This results in an equation with one unknown, which is solved by a single division. There is no separate "cross method" beyond the main property.
How to determine if a relationship in a problem is direct or inverse?
Mentally double the first quantity and observe the second. If it also doubles, the relationship is direct, and the proportion is set up "as is." If it halves, the relationship is inverse, and the second ratio needs to be inverted. Another sign: for a direct relationship, the ratio of quantities is constant; for an inverse relationship, their product is constant.
How to solve percentage problems using proportions?
Set up the proportion "part : whole = percent : 100" and find the unknown using cross-multiplication. For example, 12% of 350: we get x : 350 = 12 : 100, so 100x = 4200, and x = 42. The same method is used to find the whole given the part, and to find the percentage itself.
