Ratios and Proportions: Example Problems Worked Out

Equivalent ways of writing the ratio 3:2.
Equivalent ways of writing the ratio 3:2.

Ratios tell you the relative sizes of two quantities compared to each other. They can be of the form part-to-part or part-to-whole. For example, suppose you have a fruit basket that contains 7 apples and 12 bananas. The ratio of apples to bananas can be written in several equivalent forms:

  • 7:12
  • 7/12
  • 7 to 12

This is a part-to-part ratio since both apples and bananas are parts of the whole. The ratio of apples to total fruit in the basket is 7/19, since there are 7 + 12 = 19 pieces of fruit. This is a part to whole ratio.

If the basket of fruit had 14 apples and 24 bananas, the ratio of apples to bananas would still be 7/12. Why? Because when you write the ratio 14/24, it is a reducible fraction. Both the numerator and denominator can be divided by 2, to yield the simpler fraction (ratio) 7/12. This means knowing only the ratio of two quantities isn't enough info to determine the quantity of each group.

This article works out several examples of ratio and proportion problems so you can follow the procedure to work out similar questions. See also: More Ratio Example Problems Worked Out for extra practice.


Example 1: Unknown Part Quantities

At a produce store, the ratio of mangoes to peaches on display is 3:2. If an employee adds 20 more peaches to the display, the ratio of mangoes to peaches will be 5:4. How many mangoes are there?

In this problem, we know that for every 3 mangoes there are 2 peaches. But there could be 30 mangoes and 20 peaches, or 600 mangoes and 400 peaches, or any other pair of numbers whose ratio reduces to 3/2. If we call the number of mangoes M and the orignal number of peaches P, then we can write the equation

M/P = 3/2

Simplifying this equation by cross multiplying gives us

2M = 3P, or M = 1.5P

If 20 more peaches are added to the display, the new ratio becomes 5/4, meaning for every 5 mangoes there are 4 peaches. We still have M mangoes, but we now have P + 20 peaches. We can put this into mathematical notation as well:

M/(P + 20) = 5/4

Simplifying this equation by cross-multiplying gives us

4M = 5(P + 20), or M = 1.25P + 25.

Notice that we have two equations in two variables, M and P. We know that M = 1.5P and M = 1.25P + 25, therefore we can combine these pieces to eliminate M and solve for P:

1.5P = 1.25P + 25
1.5P - 1.25P = 25
0.25P = 25
P = 25/0.25 = 100.

So the original number of peaches is 100. Plugging this value back into the equation M = 1.5P gives us M = 150. So there are 150 mangoes on display.

Example 2: Unknown Total

A fruit basket contains only apples, bananas, and mangoes. Together, the number of bananas and mangoes is 75. The ratio of apples to bananas is 2/3, and the ratio of mangoes to apples is 9/4. How many pieces of fruit are there?

Lets use the variables A, B, and M for the number of apples, bananas, and mangoes respectively. Three equations we can write are

B + M = 75
A/B = 2/3
M/A = 9/4

This set of three equations in three unknowns can be simplified to

B + M = 75
B = 1.5A
M = 2.25A

If we take the last two equations and plug them into the first, we get

1.5A + 2.25A = 75
3.75A = 75
A = 75/3.75 = 20

So we know there are 20 apples. Working backward, we can deduce that the number of bananas is

B = 1.5*20 = 30

and that the number of mangoes is

M = 2.25*20 = 45.

Therefore, the total number of fruits is 20 + 30 + 45 = 105.

Example 3: Concentrations

Chemical Xyz is soluble in water. A scientist makes a 5% aqueous solution of Xyz. She then boils the solution so that 0.5 liters of water evaporate but none of the chemical evaporates; the concentration of the solution is now 8%. How many liters of water and how many liters of Xyz did she start with?

When we say a solution has 5% strength, it means that out of 100 parts of solution, 5 parts are Xyz and 95 parts are water. Thus, the ratio of Xyz to water is 5/95, or equivalently 1/19. (Not 1/20 as you might think at first glance!)

If we let X = the amount of Xyz and W = the amount of water originally in the solution, then we get the equation

X/W = 1/19, or W = 19X

When a solution has 8% strength, it means that out of 100 parts, 8 are Xyz and 92 are water. Thus, the ratio of chemical to water is 8/92, or equivalently 2/23. Since the amount of water at this stage is W - 0.5, we get the new ratio equation

X/(W - 0.5) = 2/23

This can be simplified to

W = 11.5X + 0.5

Since we know that W = 19X and W = 11.5X + 0.5, we can solve for X:

19X = 11.5X + 0.5
19X - 11.5X = 0.5
7.5X = 0.5
X = 0.5/7.5 = 0.0666667 liters

The amount of water she started with was W = 19*0.0666667 = 1.2666667 liters. Thus, the total amount of solution she started with was 0.0666667 + 1.2666667 = 1.3333333 liters, or equivalently 1 1/3 liters.

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Comments 4 comments

nicot 3 years ago

how do you solve: ratio of A to B is 1.3, ratio of B to C is 2.3, ratio of C to D is 3.3, what is the ratio of D to A?

calculus-geometry profile image

calculus-geometry 3 years ago from Germany Author

Hi Nicot, you can solve this problem with an elegant trick. First set up the equations

A/B = 1.3

B/C = 2.3

C/D = 3.3

Now if you multiply all these fractions together you get

(A/B)(B/C)(C/D) = 1.3*2.3*3.3 = 9.867

But notice how the left side reduces when the numerators and denominators cancel out, leaving you with

A/D = 9.867

The ratio of D to A is equivalent to D/A, and since A/D = 9.867, you just take the inverse to get the final answer. This gives you

D/A = 0.101348

3 years ago

thanks for this lesson...I have a question about solving a cake ingredient ratio problem: A baker makes dry batter mix using flour, sugar, and dried egg powder in a ratio of 4:2:1 by volume. If he wants to make 2.8 liters of cake mix, how much of each ingredient should he use?

calculus-geometry profile image

calculus-geometry 3 years ago from Germany Author

Given the parts 4, 2, and 1, the whole must be 4+2+1 = 7. That means the cake mix is 4/7 flour (F), 2/7 sugar (S), and 1/7 powdered egg (E). If you are going to make 2.8 liters, then the quantities of each ingredient are

F = (4/7)*2.8 = 1.6 liters

S = (2/7)*2.8 = 0.8 liters

E = (1/7)*2.8 = 0.4 liters

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