Ratio and Proportion

A ratio compares two quantities by division: it tells how many times one quantity contains the other. A proportion is an equation that says two ratios are equal. Ratios and proportions are everywhere: in recipes, maps and scale drawings, prices, speeds and mixtures, and they are the key to many word problems.

On this page: Ratio · Simplifying ratios · Dividing in a ratio · Compound and duplicate ratios · Proportion · Cross-multiplication · Mean, third and fourth proportional · Continued proportion · Properties of proportions · Direct and inverse proportion · Solved problems · Practice

What Is a Ratio?

The ratio of $a$ to $b$ is written ${a:b}$ and read "$a$ to $b$". Its value is the quotient ${a\div b}$, so a ratio can also be written as a fraction, $\frac{a}{b}$. The two numbers are the terms of the ratio. The first term, $a$, is called the antecedent, and the second term, $b$, is called the consequent.

Example. A bag holds 6 blue marbles and 4 orange marbles.

The 10 marbles make two equal groups, each with 3 blue and 2 orange marbles.

• The ratio of blue marbles to orange marbles is ${6:4}$, which simplifies to ${3:2}$: for every 3 blue marbles there are 2 orange ones.
• The ratio of orange marbles to blue marbles is ${4:6=2:3}$. The order matters: ${2:3}$ is not the same as ${3:2}$.
• The ratio of blue marbles to all the marbles is ${6:10=3:5}$.

The first two are part-to-part ratios. The last one is a part-to-whole ratio, and it is the same as a fraction of the whole: $\frac{3}{5}$, or 60%, of the marbles are blue.

Example. What is the ratio of 50 cm to 2 m? Change both lengths to centimeters first, because 2 m is 200 cm:

${{50\text{ cm}:200\text{ cm}}={1:4}}$

The second length is 4 times the first one. The answer ${50:2}$ would be wrong.

Equivalent Ratios and Simplifying

A ratio is in simplest form when its terms are whole numbers with no common factor except 1. To simplify a ratio of whole numbers, divide both terms by their greatest common factor, just as you reduce a fraction.

Example 1. Simplify ${12:18}$. The greatest common factor of 12 and 18 is 6:

${12:18}={(12\div 6):(18\div 6)}={2:3}$

Example 2. If the terms are fractions, multiply both terms by the least common denominator. To simplify ${\frac{1}{2}:\frac{2}{3}}$, multiply by 6:

${\frac{1}{2}:\frac{2}{3}}={\left(\frac{1}{2}\cdot 6\right):\left(\frac{2}{3}\cdot 6\right)}={3:4}$

Example 3. If the terms are decimals, multiply by 10, 100, … until both terms are whole numbers:

${0.4:1.2}={4:12}={1:3}$

Example 4. Simplify the ratio of 45 minutes to 2 hours. 2 hours is 120 minutes:

${{45:120}={3:8}}$

Comparing Ratios

To compare two ratios, compare their values. Cross-multiplying is often the quickest way: for positive numbers, ${a:b}$ is greater than ${c:d}$ exactly when ${ad\gt bc}$.

Example. Which ratio is greater, ${11:9}$ or ${44:35}$?

${{11\cdot 35}={385}}$ and ${{9\cdot 44}={396}}$

Since ${385\lt 396}$, the ratio ${44:35}$ is greater. Check with decimals: ${{11\div 9}\approx{1.222}}$ and ${{44\div 35}\approx{1.257}}$.

Adding the same number to both terms changes a ratio. ${2:3}$ and ${(2+1):(3+1)}$, which is ${3:4}$, are not equal, because $\frac{2}{3}$ is about 0.67 and $\frac{3}{4}$ is 0.75. Only multiplying or dividing both terms gives an equivalent ratio. In fact, adding the same positive number to both terms always moves the value of a ratio closer to 1: ${5:3}$ is about 1.67, but ${(5+1):(3+1)}$, which is ${6:4}$, is 1.5.

Ratios of Three or More Quantities

A ratio can compare more than two quantities. Concrete, for example, can be mixed from cement, sand and gravel in the ratio ${1:2:4}$: 1 bucket of cement for every 2 buckets of sand and 4 buckets of gravel. The rules are the same: all the terms can be multiplied or divided by the same number, so ${2:4:8}$ is the same ratio as ${1:2:4}$.

Example. ${a:b=2:3}$ and ${b:c=4:5}$. Find ${a:b:c}$.

Solution: The two ratios use different numbers for $b$, 3 and 4. Make them the same: the least common multiple of 3 and 4 is 12, so multiply the first ratio by 4 and the second one by 3:

${a:b=8:12}$ and ${b:c=12:15}$, so ${{a:b:c}={8:12:15}}$

Dividing an Amount in a Given Ratio

Example 1. Ann and Ben share $\$60$ in the ratio ${2:3}$. How much does each of them get?

Solution: There are ${{2+3}=5}$ equal parts, and one part is ${{60\div 5}={\$12}}$. Ann gets ${{2\cdot 12}={\$24}}$ and Ben gets ${{3\cdot 12}={\$36}}$.

$\$60$ is 5 equal parts of $\$12$. Ann gets 2 of them and Ben gets 3.

Check: ${{24+36}=60}$ ✓ and ${24:36=2:3}$ ✓

Example 2. How much cement, sand and gravel go into 350 kg of dry concrete mix in the ratio ${1:2:4}$?

Solution: There are ${{1+2+4}=7}$ parts, and one part is ${{350\div 7}=50}$ kg. The mix needs 50 kg of cement, ${{2\cdot 50}=100}$ kg of sand and ${{4\cdot 50}=200}$ kg of gravel.

Example 3. The ratio of boys to girls in a class is ${4:5}$, and there are 12 boys. How many girls are there?

Solution: The boys are 4 parts, so one part is ${{12\div 4}=3}$ students. The girls are 5 parts: ${{5\cdot 3}=15}$ girls. The whole class is ${{4+5}=9}$ parts, or ${{9\cdot 3}=27}$ students.

Compound, Duplicate and Inverse Ratios

Two or more ratios can be multiplied together. The result is called their compound ratio: the product of the antecedents to the product of the consequents. Squaring or cubing a ratio, taking a root of it or turning it around also gives ratios with their own names:

Name	Ratio	Example
Compound ratio of ${a:b}$ and ${c:d}$	${ac:bd}$	of ${2:3}$ and ${4:5}$ is ${8:15}$
Duplicate ratio of ${a:b}$	${a^2:b^2}$	of ${2:3}$ is ${4:9}$
Triplicate ratio of ${a:b}$	${a^3:b^3}$	of ${2:3}$ is ${8:27}$
Sub-duplicate ratio of ${a:b}$	${\sqrt{a}:\sqrt{b}}$	of ${4:9}$ is ${2:3}$
Sub-triplicate ratio of ${a:b}$	${\sqrt[3]{a}:\sqrt[3]{b}}$	of ${8:27}$ is ${2:3}$
Inverse (reciprocal) ratio of ${a:b}$	${b:a}$, which equals ${\frac{1}{a}:\frac{1}{b}}$	of ${2:3}$ is ${3:2}$

Example. Find the ratio compounded of ${2:3}$, the duplicate ratio of ${3:4}$ and the sub-duplicate ratio of ${64:9}$.

Solution: The duplicate ratio of ${3:4}$ is ${9:16}$, and the sub-duplicate ratio of ${64:9}$ is ${8:3}$. Multiply the antecedents and the consequents:

${(2\cdot 9\cdot 8):(3\cdot 16\cdot 3)}={144:144}={1:1}$

Duplicate is not double. The duplicate ratio of ${3:1}$ is ${9:1}$, the ratio squared, while twice the ratio is ${6:1}$. In the same way, the triplicate ratio of ${3:1}$ is ${27:1}$, not ${9:1}$.

When the consequent of each ratio is the antecedent of the next one, the middle terms cancel. The ratio compounded of ${a:b}$, ${b:c}$ and ${c:d}$ is ${a:d}$, because

$$\frac{a}{b}\cdot\frac{b}{c}\cdot\frac{c}{d}=\frac{a}{d}$$

This fact is used for continued proportions below.

What Is a Proportion?

A proportion is an equation that says two ratios are equal:

$$a:b=c:d\qquad\text{or}\qquad\frac{a}{b}=\frac{c}{d}$$

It is read "$a$ is to $b$ as $c$ is to $d$", and old books write it as ${a:b::c:d}$. The numbers $a$, $b$, $c$ and $d$ are the terms of the proportion. The outer terms $a$ and $d$ are the extremes, and the inner terms $b$ and $c$ are the means. Four numbers that form a proportion are said to be proportional, or in proportion.

The extremes are the outer terms, and the means are the inner terms.

Example. Are 12, 8, 15 and 10 in proportion? The values of the two ratios are ${{12\div 8}={1.5}}$ and ${{15\div 10}={1.5}}$. They are equal, so ${12:8=15:10}$ is a proportion.

In everyday speech, "proportion" often means a part or a share, as in "a large proportion of the students". In mathematics a proportion is always an equality of two ratios.

The Main Property: Cross-Multiplication

To see why, multiply both sides of ${\frac{a}{b}=\frac{c}{d}}$ by $bd$:

${{\frac{a}{b}\cdot bd}={\frac{c}{d}\cdot bd}}$, which simplifies to ${ad=bc}$.

The rule is called cross-multiplication, because the two products go across the equals sign, from corner to corner. It gives a quick test for a proportion:

${12:8=15:10}$ is a proportion, because ${{12\cdot 10}={120}}$ and ${{8\cdot 15}={120}}$.

${3:4}$ and ${5:7}$ are not in proportion: ${{3\cdot 7}={21}}$, but ${{4\cdot 5}={20}}$.

It works the other way round too: if ${ad=bc}$, the four numbers make a proportion. The factors on one side of the equation become the extremes, and the factors on the other side become the means. For example, ${3\cdot 8=4\cdot 6}$ gives the proportions ${3:4=6:8}$ and ${3:6=4:8}$.

Solving a Proportion

If one term of a proportion is unknown, cross-multiply and solve the equation.

Example 1. Solve ${x:12=5:4}$.

${4x}={12\cdot 5}={60}$, so ${x=15}$

Check: ${{15\div 12}={1.25}}$ and ${{5\div 4}={1.25}}$ ✓

Example 2. Solve ${\frac{x+2}{3}=\frac{x-1}{2}}$.

${{2(x+2)}={3(x-1)}}$

${{2x+4}={3x-3}}$, so ${x=7}$

Check: ${{\frac{7+2}{3}}={3}}$ and ${{\frac{7-1}{2}}={3}}$ ✓

Solving ${ad=bc}$ for each letter shows how to find any term from the other three. An extreme is the product of the means divided by the other extreme, and a mean is the product of the extremes divided by the other mean:

$$a=\frac{bc}{d}\qquad d=\frac{bc}{a}$$ $$b=\frac{ad}{c}\qquad c=\frac{ad}{b}$$

Fourth, Third and Mean Proportional

The fourth proportional to $a$, $b$ and $c$ is the number $x$ for which ${a:b=c:x}$. Cross-multiplying gives ${ax=bc}$, so

$$x=\frac{bc}{a}$$

Example. The fourth proportional to 4, 6 and 10 is ${{\frac{6\cdot 10}{4}}={15}}$, because ${4:6=10:15}$.

This is the old rule of three: three numbers are given, and the fourth one is found by multiplying the second and the third and dividing by the first.

The third proportional to $a$ and $b$ is the number $x$ for which ${a:b=b:x}$. Then ${ax=b^2}$, so

$$x=\frac{b^2}{a}$$

Example. The third proportional to 4 and 6 is ${{\frac{6^2}{4}}={9}}$, because ${4:6=6:9}$.

The mean proportional between two positive numbers $a$ and $b$ is the positive number $x$ for which ${a:x=x:b}$. Then ${x^2=ab}$, so

$$x=\sqrt{ab}$$

The mean proportional is also called the geometric mean of $a$ and $b$.

Example. The mean proportional between 2 and 18 is ${\sqrt{2\cdot 18}}={\sqrt{36}}={6}$, because ${2:6=6:18}$.

In the proportion ${4:6=6:9}$, the number 6 is the mean proportional between 4 and 9, and 9 is the third proportional to 4 and 6.

Continued Proportion

Three or more numbers are in continued proportion when the ratio of each number to the next one is the same:

$$a:b=b:c=c:d=\ldots$$

For example, 64, 32, 16, 8 and 4 are in continued proportion, because ${64:32}$, ${32:16}$, ${16:8}$ and ${8:4}$ all equal 2. Each number is the one before it multiplied by the same number, here by $\frac{1}{2}$, so numbers in continued proportion form a geometric progression.

Example 1. Are 4, 10 and 25 in continued proportion? ${{10^2}={100}}$ and ${{4\cdot 25}={100}}$, so yes: ${4:10=10:25}$.

The First Number to the Last

If $a$, $b$, $c$ are in continued proportion, then ${{\frac{a}{c}}={\frac{a}{b}\cdot\frac{b}{c}}}$, and both factors are equal to $\frac{a}{b}$. So

$$a:c=a^2:b^2$$

In words, the first number is to the third in the duplicate ratio of the first to the second. For four numbers in continued proportion, ${a:d=a^3:b^3}$, and so on: the ratio of the first number to the last one is the common ratio raised to a power one less than the number of terms. In 64, 32, 16, 8, 4 the common ratio is 2 and there are 5 numbers, so ${64:4}$ is ${2^4=16}$.

Example 2. The numbers 2, $x$, $y$, 54 are in continued proportion. Find $x$ and $y$.

Solution: Each number is the one before it multiplied by the same number $q$, so ${2q^3=54}$. Then ${q^3=27}$ and ${q=3}$: the numbers are 2, 6, 18 and 54, so ${x=6}$ and ${y=18}$.

Properties of Proportions

A proportion can be rearranged or combined in many ways and still be a proportion. The rules have Latin names that many textbooks still use. In the table, ${a:b=c:d}$ and all the terms are nonzero. Each example starts from ${12:4=6:2}$, where both ratios equal 3.

Name	Rule	Example
Invertendo (inversion)	${{b:a}={d:c}}$	${{4:12}={2:6}}$
Alternendo or alternando (alternation)	${{a:c}={b:d}}$	${{12:6}={4:2}}$
Componendo (composition)	${{(a+b):b}={(c+d):d}}$	${{16:4}={8:2}}$
Dividendo (division)	${{(a-b):b}={(c-d):d}}$	${{8:4}={4:2}}$
Componendo and dividendo	${(a+b):(a-b)}={(c+d):(c-d)}$	${{16:8}={8:4}}$
Convertendo (conversion)	${{a:(a-b)}={c:(c-d)}}$	${{12:8}={6:4}}$
Addendo (sum of antecedents to sum of consequents)	${{(a+c):(b+d)}={a:b}}$	${{18:6}={12:4}}$
Multiplying both antecedents (or both consequents) by the same number	${{ka:b}={kc:d}}$	${{24:4}={12:2}}$
Powers and roots	${{a^n:b^n}={c^n:d^n}}$	${{144:16}={36:4}}$
Multiplying two proportions	if also ${{e:f}={g:h}}$, then ${{ae:bf}={cg:dh}}$	with ${{10:5}={8:4}}$: ${{120:20}={48:8}}$

Componendo and dividendo and convertendo also need ${a\ne b}$, so that no term is zero.

Why do these rules hold? Call the common value of the two ratios $k$, so that ${\frac{a}{b}=\frac{c}{d}=k}$. Then ${a=bk}$ and ${c=dk}$. Put these into both sides of a rule. For componendo and dividendo:

${\frac{a+b}{a-b}}={\frac{bk+b}{bk-b}}={\frac{k+1}{k-1}}$

${\frac{c+d}{c-d}}={\frac{dk+d}{dk-d}}={\frac{k+1}{k-1}}$

Both sides equal the same number, so they are equal. This substitution, often called the $k$-method, proves the other rules too, and it solves many "prove that" problems.

Example. If ${\frac{a+b}{a-b}=\frac{7}{3}}$, find ${a:b}$.

Solution: By componendo and dividendo,

${\frac{(a+b)+(a-b)}{(a+b)-(a-b)}}={\frac{7+3}{7-3}}$

${{\frac{2a}{2b}}={\frac{10}{4}}}$, so ${a:b=5:2}$

Check: with ${a=5}$ and ${b=2}$, ${{\frac{5+2}{5-2}}={\frac{7}{3}}}$ ✓

Direct and Inverse Proportion

Two quantities that change together are directly proportional if their ratio always stays the same: when one of them is doubled, so is the other, and when one is divided by 3, so is the other. If $y$ is directly proportional to $x$, then

$$\frac{y}{x}=k\qquad\text{or}\qquad y=kx$$

The fixed number $k$ is called the constant of proportionality.

Two quantities are inversely proportional if their product always stays the same: when one of them is doubled, the other is halved. Then

$$xy=k\qquad\text{or}\qquad y=\frac{k}{x}$$

A direct proportion is a straight line through the origin. An inverse proportion is a curve called a hyperbola.

	Direct proportion	Inverse proportion
When $x$ is doubled	$y$ is doubled	$y$ is halved
What stays the same	the ratio $\frac{y}{x}$	the product $xy$
Formula	${y=kx}$	${y=\frac{k}{x}}$
Two pairs of values	${{x_1:x_2}={y_1:y_2}}$	${{x_1:x_2}={y_2:y_1}}$
Examples	the number of items and their total price; the time and the distance at a steady speed	the number of workers and the time a job takes; the speed and the time for a fixed distance

Example 1. 3 kg of apples cost $\$7.50$. How much do 5 kg cost?

Solution: The price is directly proportional to the weight. With the unitary method, first find the price of 1 kg: ${{7.50\div 3}={\$2.50}}$. Then 5 kg cost ${{5\cdot 2.50}={\$12.50}}$. With a proportion, the ratio of the weights equals the ratio of the prices:

${3:5=7.50:x}$, so ${{3x}={37.50}}$ and ${{x}={\$12.50}}$

Example 2. 6 workers can paint a fence in 10 hours. How long would 4 workers need, if they all work at the same rate?

Solution: Fewer workers need more time. The number of workers and the time are inversely proportional, so their product stays the same. The job takes ${{6\cdot 10}=60}$ worker-hours, and 4 workers need ${{60\div 4}=15}$ hours. With a proportion, the ratio of the numbers of workers equals the inverse ratio of the times:

${6:4=x:10}$, so ${{4x}={60}}$ and ${x=15}$ hours

Example 3. At 60 km/h a trip takes 3 hours. How long does it take at 90 km/h? The distance is ${{60\cdot 3}=180}$ km, so at 90 km/h the trip takes ${{180\div 90}=2}$ hours.

Not everything that grows together is proportional. A taxi ride costs a $\$3$ fixed fee plus $\$2$ per kilometer. Then 1 km costs $\$5$ and 2 km cost $\$7$, not $\$10$. The price grows with the distance, but it is not directly proportional to it, because the fixed fee doesn't change: ${5:1}$ is not equal to ${7:2}$.

Common Mistakes

Wrong	Right
✗ 3 boys and 5 girls: the ratio of boys to girls is ${5:3}$	✓ Keep the order of the words: boys to girls is ${3:5}$
✗ ${50\text{ cm}:2\text{ m}}={25:1}$	✓ Use the same units: ${50\text{ cm}:200\text{ cm}}={1:4}$
✗ $\$60$ in the ratio ${2:3}$ is ${60\div 2=\$30}$ and ${60\div 3=\$20}$	✓ 5 parts of $\$12$: $\$24$ and $\$36$
✗ ${2:3}={(2+1):(3+1)}={3:4}$	✓ Multiply or divide both terms: ${{2:3}={4:6}}$
✗ The duplicate ratio of ${3:1}$ is ${6:1}$	✓ It is ${{3^2:1^2}={9:1}}$
✗ ${\frac{x}{2}+1=\frac{5}{4}}$, so ${4x+1=10}$	✓ First make each side a single fraction: ${\frac{x+2}{2}=\frac{5}{4}}$, so ${4(x+2)=10}$ and ${x=0.5}$
✗ 6 workers need 10 hours, so 4 workers need ${\frac{4\cdot 10}{6}\approx 6.7}$ hours	✓ Fewer workers need more time: ${\frac{6\cdot 10}{4}=15}$ hours
✗ The mean proportional between 4 and 9 is ${\frac{4+9}{2}=6.5}$	✓ It is ${\sqrt{4\cdot 9}=6}$, because ${{4:6}={6:9}}$

Solved Problems

Problem 1. Divide 49 into two parts so that the greater part increased by 6 is to the smaller part decreased by 11 as 9 is to 2.

Solution: Let the greater part be $x$. Then the smaller part is ${49-x}$, and decreased by 11 it is ${38-x}$:

${{(x+6):(38-x)}={9:2}}$

${{2(x+6)}={9(38-x)}}$

${{2x+12}={342-9x}}$, so ${11x=330}$ and ${x=30}$

The parts are 30 and 19. Check: ${36:8=9:2}$ ✓

Problem 2. What number must be added to each of 1, 5 and 13 so that the three sums are in continued proportion?

Solution: Let the number be $x$. The sums ${x+1}$, ${x+5}$ and ${x+13}$ are in continued proportion when the square of the middle one equals the product of the other two:

${(x+5)^2}={(x+1)(x+13)}$

${x^2+10x+25}={x^2+14x+13}$, so ${4x=12}$ and ${x=3}$

The sums are 4, 8 and 16, and ${4:8=8:16}$ ✓

Problem 3. Divide 18 into two parts whose squares are in the ratio ${25:16}$.

Solution: If the squares of the parts are in the ratio ${25:16}$, the parts themselves are in the sub-duplicate ratio, ${{\sqrt{25}:\sqrt{16}}={5:4}}$. Dividing 18 in the ratio ${5:4}$ gives 9 parts of 2, so the parts are 10 and 8.

Check: ${100:64=25:16}$ ✓

Problem 4. The number 20 is divided into two parts that are in the duplicate ratio of ${3:1}$. Find the mean proportional between the two parts.

Solution: The duplicate ratio of ${3:1}$ is ${9:1}$. Dividing 20 in the ratio ${9:1}$ gives 10 parts of 2, so the parts are 18 and 2. Their mean proportional is

${\sqrt{18\cdot 2}}={\sqrt{36}}={6}$

Problem 5. Two numbers are in the ratio ${3:2}$. If 6 is added to the greater number and 6 is subtracted from the smaller one, the results are in the ratio ${3:1}$. Find the numbers.

Solution: Write the numbers as $3k$ and $2k$:

${(3k+6):(2k-6)}={3:1}$

${3k+6}={3(2k-6)}={6k-18}$, so ${3k=24}$ and ${k=8}$

The numbers are 24 and 16. Check: ${30:10=3:1}$ ✓

Problem 6. Two positive numbers are in the duplicate ratio of ${4:3}$, and 24 is their mean proportional. Find the numbers.

Solution: The duplicate ratio of ${4:3}$ is ${16:9}$, so write the numbers as $16k$ and $9k$ with ${k\gt 0}$. Their product is the square of the mean proportional:

${{16k\cdot 9k}={24^2}}$, so ${144k^2=576}$, ${k^2=4}$ and ${k=2}$

The numbers are 32 and 18. Check: ${32:24}$ and ${24:18}$ both equal $\frac{4}{3}$ ✓

Problem 7. The product of two numbers is 135, and the difference of their squares is to the square of their difference as 4 is to 1. Find the numbers.

Solution: Call the numbers $a$ and $b$. Since ${a^2-b^2}={(a+b)(a-b)}$, the condition says

${\frac{(a+b)(a-b)}{(a-b)^2}}={\frac{a+b}{a-b}}={\frac{4}{1}}$

By componendo and dividendo, ${{\frac{2a}{2b}}={\frac{4+1}{4-1}}}$, so ${a:b=5:3}$. Write ${a=5k}$ and ${b=3k}$:

${{5k\cdot 3k}={135}}$, so ${k^2=9}$ and ${k=\pm 3}$

The numbers are 15 and 9, or −15 and −9. Check: ${{15\cdot 9}={135}}$, and ${(225-81):36}={144:36}={4:1}$ ✓

Problem 8. Solve ${\frac{\sqrt{x+5}+\sqrt{x-5}}{\sqrt{x+5}-\sqrt{x-5}}=3}$.

Solution: Write 3 as $\frac{3}{1}$ and use componendo and dividendo. The sum and the difference of the two roots turn into single roots:

${\frac{2\sqrt{x+5}}{2\sqrt{x-5}}}={\frac{3+1}{3-1}}={2}$

${{\sqrt{x+5}}={2\sqrt{x-5}}}$, so ${{x+5}={4(x-5)}}$, ${3x=25}$ and ${x=\frac{25}{3}}$

Check: ${{x+5}={\frac{40}{3}}}$ is 4 times ${{x-5}={\frac{10}{3}}}$, so ${{\sqrt{x+5}}={2\sqrt{x-5}}}$, and the left side is ${{\frac{2+1}{2-1}}={3}}$ ✓

Practice

Try each problem, then open it to check your answer.

1. Simplify ${24:36}$.

The greatest common factor is 12: ${{24:36}={2:3}}$

2. Simplify ${\frac{3}{4}:\frac{5}{6}}$.

Multiply both terms by 12: ${{\frac{3}{4}:\frac{5}{6}}={9:10}}$

3. Which ratio is greater, ${7:9}$ or ${10:13}$?

${{7\cdot 13}={91}}$ and ${{9\cdot 10}={90}}$. Since ${91\gt 90}$, the ratio ${7:9}$ is greater.

4. Divide $\$96$ in the ratio ${3:5}$.

There are 8 parts of $\$12$: $\$36$ and $\$60$.

5. Two numbers are in the ratio ${5:3}$, and their difference is 14. Find the numbers.

The difference is ${{5-3}=2}$ parts, so one part is 7. The numbers are 35 and 21.

6. ${a:b=3:4}$ and ${b:c=6:7}$. Find ${a:b:c}$.

Make $b$ equal to 12 in both ratios: ${a:b=9:12}$ and ${b:c=12:14}$, so ${{a:b:c}={9:12:14}}$.

7. Solve ${(x-1):6=5:3}$.

${{3(x-1)}={30}}$, so ${x-1=10}$ and ${x=11}$.

8. Find the third proportional to 9 and 12, and the mean proportional between 8 and 18.

The third proportional is ${{\frac{12^2}{9}}={16}}$, because ${9:12=12:16}$. The mean proportional is ${{\sqrt{8\cdot 18}}={12}}$, because ${8:12=12:18}$.

9. Find the ratio compounded of ${7:5}$, the duplicate ratio of ${4:9}$ and the triplicate ratio of ${3:2}$.

${(7\cdot 16\cdot 27):(5\cdot 81\cdot 8)}={3024:3240}={14:15}$

10. If ${(a+2b):(a-2b)=9:5}$, find ${a:b}$.

By componendo and dividendo, ${{\frac{2a}{4b}}={\frac{14}{4}}}$, so ${\frac{a}{b}=7}$ and ${a:b=7:1}$.

11. If ${a:b=c:d}$, prove that ${(a^2+b^2):(c^2+d^2)=ab:cd}$.

Put ${a=bk}$ and ${c=dk}$. The left side is ${\frac{b^2k^2+b^2}{d^2k^2+d^2}}={\frac{b^2(k^2+1)}{d^2(k^2+1)}}={\frac{b^2}{d^2}}$, and the right side is ${\frac{bk\cdot b}{dk\cdot d}}={\frac{b^2}{d^2}}$. The two sides are equal.

12. A map has the scale ${1:50{,}000}$. Two villages are 7.4 cm apart on the map. How far apart are they in reality?

${7.4\cdot 50{,}000}={370{,}000}$ cm, which is 3,700 m or 3.7 km.

13. 4 pipes fill a tank in 9 hours. How long do 6 such pipes take?

The number of pipes and the time are inversely proportional: ${{4\cdot 9}={36}}$, and ${{36\div 6}={6}}$ hours.

More practice, with new numbers every time: which proportion gives the price?