Mathematics
Proportions: 8 Solved Exercises and the 4 Properties
In a proportion the product of the means equals the product of the extremes: that one rule finds any missing term. 8 solved exercises, tables and typical mistakes.
Recommended for: Grade 6 · Grade 7 · Grade 8
A proportion is an equality between two ratios: 3 : 12 = 5 : 20 reads “3 is to 12 as 5 is to 20”, and it is true because both ratios equal 0.25. One single rule solves every exercise: the product of the means equals the product of the extremes.
The names of the terms
In a proportion a : b = c : d each of the four numbers has a precise name, and mixing them up is the mistake that derails everything else.
| Term | Position in a : b = c : d | Name |
|---|---|---|
| a | first | antecedent of the first ratio, extreme |
| b | second | consequent of the first ratio, mean |
| c | third | antecedent of the second ratio, mean |
| d | fourth | consequent of the second ratio, extreme |
The extremes are the first and the last term, the means are the two in the middle. Hence the fundamental property: b · c = a · d. In many textbooks the same rule is called cross multiplication, because a : b = c : d can be written as the fractions a/b = c/d and you multiply across the equals sign.
How to find the missing term
There are no four separate formulas to memorise: the mechanism is always the same, only the pair you multiply changes.
| Unknown | Formula | Example |
|---|---|---|
| first term (extreme) | x = (b · c) : d | x : 12 = 5 : 20 → x = (12 · 5) : 20 = 3 |
| second term (mean) | x = (a · d) : c | 3 : x = 5 : 20 → x = (3 · 20) : 5 = 12 |
| third term (mean) | x = (a · d) : b | 3 : 12 = x : 20 → x = (3 · 20) : 12 = 5 |
| fourth term (extreme) | x = (b · c) : a | 3 : 12 = 5 : x → x = (12 · 5) : 3 = 20 |
In practice: multiply the two known terms of the type opposite to the unknown, then divide by the third known term.
The four properties worth remembering
Besides the fundamental property there are four transformations that keep a proportion true. They matter above all in problems that give you the sum or the difference of the terms.
| Property | From a : b = c : d you get | When to use it |
|---|---|---|
| inverting | b : a = d : c | to move the unknown to the numerator |
| swapping the means | a : c = b : d | to compare quantities of the same kind |
| adding (componendo) | (a + b) : b = (c + d) : d | when the problem gives the sum |
| subtracting (dividendo) | (a − b) : b = (c − d) : d | when the problem gives the difference |
An example of componendo: if two shares are in the ratio 3 : 4, adding gives (3 + 4) : 4 = 7 : 4. If the two shares add up to 84, write 84 : x = 7 : 4 and the larger share follows immediately, x = (84 · 4) : 7 = 48. That is the same result as exercise four, where the total is split into 7 equal parts.
Directly and inversely proportional quantities
Half of all mistakes happen here, because the proportion pattern seems to work every time, but the way you write it changes.
| Directly proportional | Inversely proportional | |
|---|---|---|
| If one doubles | the other doubles | the other halves |
| What stays constant | the ratio y : x | the product x · y |
| Formula | y = k · x | y = k : x |
| Examples | notebooks and cost, hours worked and pay, recipe ingredients | workers and days, speed and travel time, number of people and how long supplies last |
With inversely proportional quantities the proportion has to be written inverting one of the two ratios: 6 workers is to 4 workers as the days needed by 4 is to the days needed by 6.
The most common mistakes
The first is writing a direct proportion where the link is inverse: if you reduce the number of workers and end up with a shorter time, the answer is wrong no matter how clean the arithmetic looks. Before calculating, always ask whether the result should be larger or smaller than the starting figure.
The second is mixing units: in a proportion between lengths you cannot have centimetres on one side and metres on the other. Convert everything before applying the fundamental property.
The third concerns areas. If you enlarge a figure by a factor of 1.6, lengths are multiplied by 1.6 but areas by 1.6² = 2.56, and volumes by 1.6³. Areas are not proportional to the sides, but to their squares.
The fourth is purely arithmetic: getting the final division wrong. A ratio is a fraction, so it pays to simplify before multiplying, exactly as in operations with fractions.
Proportions, percentages and equations
A percentage is a proportion with one term fixed at 100: “15% of 240” becomes 15 : 100 = x : 240. If that pattern suits you better, you can solve every exercise on percentages with proportions, discounts and mark-ups included.
Applying the fundamental property, moreover, always turns a proportion into an equality of the form a · x = b: from there on you are solving a linear equation, which is why the last step is always dividing both sides by the coefficient of x.
How to work through the exercises
The eight exercises below follow the order in which the topic is really learned: first finding a missing term and checking a proportion, then direct proportionality problems, sharing a total in a given ratio and map scales, then the inverse case that catches most students out, and finally the geometric mean and the scale enlargement, where the ratio of the areas is not the ratio of the sides. Every answer is checked by working the calculation backwards.
Solved exercises
1. Find the missing term: 3 : 12 = 5 : x. base
Show solution
- Identify the terms: 3 and x are the extremes, 12 and 5 are the means.
- Apply the fundamental property (cross multiplication): 3 · x = 12 · 5.
- Work out the right-hand side: 3 · x = 60.
- Divide both sides by 3: x = 60 : 3 = 20.
- Check: 3 : 12 = 0.25 and 5 : 20 = 0.25. The two ratios match.
Answer: x = 20
2. Check whether 8 : 12 = 14 : 21 is a true proportion. base
Show solution
- You do not need the divisions: just compare the two products.
- Product of the extremes: 8 · 21 = 168.
- Product of the means: 12 · 14 = 168.
- The products are equal, so the equality is a proportion.
- Cross-check with simplified fractions: 8/12 = 2/3 and 14/21 = 2/3.
Answer: Yes, it is a proportion (168 = 168)
3. 4 notebooks cost $6.00. How much do 7 notebooks of the same kind cost? base
Show solution
- Number of notebooks and total cost are directly proportional: twice the notebooks, twice the cost.
- Write the proportion keeping the same order: 4 : 6 = 7 : x.
- Apply the fundamental property: 4 · x = 6 · 7 = 42.
- Divide by 4: x = 42 : 4 = 10.5.
- Check with the unit price: 6 : 4 = $1.50 per notebook, and 1.50 · 7 = $10.50.
Answer: $10.50
4. Share 84 candies between Emma and Noah so that their parts are in the ratio 3 : 4. intermedio
Show solution
- The ratio 3 : 4 means the total is split into 3 + 4 = 7 equal parts.
- Find the value of one part: 84 : 7 = 12 candies.
- Emma's share: 3 · 12 = 36 candies.
- Noah's share: 4 · 12 = 48 candies.
- Check: 36 + 48 = 84 and 36 : 48 = 3 : 4.
Answer: Emma 36, Noah 48
5. On a map with scale 1 : 200,000 the distance between two towns is 7.5 cm. What is the real distance in kilometres? intermedio
Show solution
- A scale of 1 : 200,000 means 1 cm on the map stands for 200,000 cm in reality.
- Write the proportion: 1 : 200,000 = 7.5 : x.
- Apply the fundamental property: x = 7.5 · 200,000 = 1,500,000 cm.
- Convert: 1,500,000 cm = 15,000 m = 15 km.
- Check: 15 km is 1,500,000 cm, and 1,500,000 : 200,000 = 7.5 cm on the map.
Answer: 15 km
6. 6 workers finish a job in 10 days. How many days do 4 workers need at the same pace? intermedio
Show solution
- Careful: these quantities are inversely proportional, because fewer workers means more days.
- The product workers × days stays constant: 6 · 10 = 60 worker-days in total.
- Divide the total work by the new number of workers: x = 60 : 4 = 15 days.
- As a proportion you must invert one ratio: 6 : 4 = x : 10.
- Check: 4 · 15 = 60, exactly like 6 · 10. Using the direct proportion 6 : 10 = 4 : x would give about 6.7 days, that is fewer workers finishing sooner: impossible.
Answer: 15 days
7. Find the geometric mean x in the continued proportion 4 : x = x : 25. avanzato
Show solution
- In a continued proportion the two means are the same number: here both are x.
- Apply the fundamental property: x · x = 4 · 25, that is x² = 100.
- Take the square root: x = 10 (positive value, since these are measures).
- Check: 4 : 10 = 0.4 and 10 : 25 = 0.4.
- General rule: the geometric mean of two numbers is the square root of their product.
Answer: x = 10
8. A photo is 15 cm wide and 10 cm tall. It is enlarged so that the width becomes 24 cm, without distortion. What is the new height, and by how much does the area grow? avanzato
Show solution
- Without distortion the sides are proportional: 15 : 10 = 24 : x.
- Apply the fundamental property: 15 · x = 10 · 24 = 240, so x = 240 : 15 = 16 cm.
- Find the scale factor: 24 : 15 = 1.6 (and indeed 16 : 10 = 1.6).
- Starting area: 15 · 10 = 150 cm². Final area: 24 · 16 = 384 cm².
- Ratio of the areas: 384 : 150 = 2.56 = 1.6². Lengths grow by a factor of 1.6, areas by 1.6².
Answer: Height 16 cm; the area becomes 2.56 times the original
FAQ
What is the fundamental property of proportions?
In any proportion a : b = c : d the product of the means equals the product of the extremes, that is b · c = a · d. It is the rule you use to find any missing term, and also to check whether four numbers really form a proportion.
How do you find the missing term in a proportion?
If the unknown is an extreme, multiply the two means and divide by the known extreme. If the unknown is a mean, multiply the two extremes and divide by the known mean. In short: multiply the two terms of the opposite type to the unknown and divide by the third known term.
How do I know if two quantities are directly or inversely proportional?
Ask what happens if you double the first quantity. If the second one doubles too they are directly proportional and their ratio is constant (notebooks and cost). If the second one halves they are inversely proportional and their product stays constant (workers and days).
What is the geometric mean used for?
It appears in continued proportions of the form a : x = x : b, where x is the square root of a · b. In geometry it underpins the right-triangle altitude theorem, in which the altitude is the geometric mean of the two segments of the hypotenuse.
How do you solve map scale problems?
A scale of 1 : 200,000 means 1 cm on the map corresponds to 200,000 cm in reality. Multiply the map measurement by the denominator of the scale, then convert: 100,000 cm make 1 km, so divide the centimetres by 100,000.