The word problems on the proportion aid you in a mathematical comparison between two numbers. The proportion gets majorly based on ratio and fractions. This article on Proportion Word Problems Worksheet with Answers PDF gives you a various number of proportion problems. At the end of this page, you can get clear knowledge of the concept of proportion. It says when two ratios are equivalent they are in proportion.
Proportion encourages solving many real-life problems. The ratio and proportion are key foundations to grasp the various concepts in mathematics. Proportions are denoted by the symbol “::”, “=”. This free printable Worksheet on Word Problems on Proportion provides various types of questions with answers to make you understand clearly how to solve the proportions problems in exams. By solving proportion examples with answers pdf, you can also improve your problem-solving skills & math skills.
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Proportion Word Problems with Answers
Example 1:
If the numbers are 4, 15, 7, and 20. What number should be added to make the numbers proportional?
Solution:
Given numbers are 4, 15, 7, and 20.
Let the needed number be k.
Now, we write the numbers according to the problem
4+k, 15+k, 7+k, and 20+k are proportional numbers.
Here,
\(\frac{4+k}{15+k}\) = \(\frac{7+k}{20+k}\)
⇒ (4+k) × (20+k) = (7+k) × (15+k)
⇒ 80+ 4k +20k +k² = 105+ 7k+ 15k+ k²
⇒ 24k+80 = 22k+42
⇒ 24k- 22k = 105-80
⇒ 2k = 25
⇒ k = \(\frac{25}{2}\)
⇒ k = 12.5
Thus, the required number is 12.5.
Example 2:
Priya enlarged the size of a photo to a height of 20 inches. What is the new width if it was originally 4 inches tall and 2 inches in width?
Solution:
The height of a photo is 20 inches.
Now, to find the new width.
The ratio to calculate is 4: 2 = 20: w
⇒ \(\frac{4}{2}\) = \(\frac{20}{w}\)
⇒ 4w = 20×2
⇒ 4w = 40
⇒ w = \(\frac{40}{4}\)
⇒ w = 10.
Thus, the new width of the photo is 10inches.
Example 3:
If P is directly proportional to n when P is 4 and n is 6. Find the value of P when n is 8.
Solution:
Given P is proportional to n ( P ∝ n ).
Now, convert to an equation multiplied by k the constant of variation.
⇒ P = nk
To find k use the given condition
P = 4 when n = 6
P = nk
⇒ k = \(\frac{n}{P}\) = \(\frac{6}{4}\)
Now, the equation is P = \(\frac{6}{4}\) × n = \(\frac{6n}{4}\)
When n = 8, then
P= \(\frac{6n}{4}\) = \(\frac{6×8}{4}\) = \(\frac{48}{4}\) = 12.
Hence, the value of P is 12 when n is 8.
Example 4:
Find the third proportion of 14 and 22?
Solution:
The proportions given are 14 and 22.
Let the third proportion be x.
According to the problem, 14 and 22 are in continued proportion.
Now,
\(\frac{14}{22}\) = \(\frac{22}{x}\)
⇒ 14× x = 22 × 22
⇒ 14x = 484
⇒ x = 484/14 = 34.57
Therefore, the third proportion is 34.57.
Example 5:
Ram, Prudhvi, and Mahesh have $15, $22, and $25 respectively with them. Father asks them to give him an equal amount so that the money held by them now is in continued proportion. Find the amount taken from each of them?
Solution:
Let the amount taken from each of them be $m.
Now,
15-m, 22-m, and 25-m are in continued proportion.
Thus, \(\frac{15-m}{22-m}\) = \(\frac{22-m}{25-m}\)
⇒ (15-m)(25-m) = (22-m)(22-m)
⇒ 375 – 15m -25m +m² = 484 – 44m +m²
⇒ 375 – 40m = 484 -44m
⇒ 375 – 484 = -44m +40m
⇒ -109 = -4m
⇒ m = 109/4
⇒ m = 27.25
Therefore, the required amount is $27.25.
Example 6:
Keerti ran 150meters in 25seconds. How long did she take to run 3meters?
Solution:
Given that
keerti ran 150metres in 25seconds.
Now, to find the time she takes to run 3mts.
Let k be the time required.
\(\frac{25}{150}\) = \(\frac{k}{3}\)
⇒ k = \(\frac{25}{150}\) × 3
⇒ k = \(\frac{75}{150}\)
⇒ k = 0.5
Thus, keerti took 0.5seconds to complete 3meters.
Example 7:
Check whether the following numbers form a proportion or not.
(i) 4.5, 3.5, 6.6, and 8.8
(ii) 2\(\frac{2}{4}\), 1\(\frac{3}{2}\), 2.2, and 5.5
Solution:
(i) Given 4.5: 3.5 = \(\frac{4.5}{3.5}\) = \(\frac{45}{35}\) = \(\frac{9}{7}\)
6.6: 8.8 = \(\frac{6.6}{8.8}\) = \(\frac{66}{88}\) = \(\frac{6}{8}\)
Therefore, \(\frac{4.5}{3.5}\) ≠ \(\frac{6.6}{8.8}\)
Thus, 4.5, 3.5, 6.6, and 8.8 are not in proportion.
(ii) Given 2\(\frac{2}{4}\), 1\(\frac{3}{2}\), 2.2, and 5.5
2\(\frac{2}{4}\): 1\(\frac{3}{2}\)
= \(\frac{10}{4}\): \(\frac{5}{2}\)
= \(\frac{10}{4}\) × 4: \(\frac{5}{2}\) × 4
= 10: 10 = 1: 1
2.2: 5.5 = \(\frac{2.2}{5.5}\) = \(\frac{22}{55}\) = \(\frac{2}{5}\) = 2: 5
Thus, the given numbers 2\(\frac{2}{4}\), 1\(\frac{3}{2}\), 2.2, and 5.5 are not in proportion.
Example 8:
Find the fourth proportional of numbers 3, 6, and 12?
Solution:
Given three proportional numbers are 5, 6, and 12.
Let the fourth proportional be x.
Now, 3, 6,12, and x be the proportionality numbers.
Thus,
\(\frac{3}{6}\) = \(\frac{12}{x}\)
⇒ 3x = 12×6
⇒ 3x = 72
⇒ x = \(\frac{72}{3}\)
⇒ x = 24.
Hence, the fourth proportional number is 24.
Example 9:
If the mean proportion is 16 and the third proportional is 64 then find the two numbers?
Solution:
Given mean proportion is 16 and the third proportion is 64.
Let the required numbers be a and b.
According to the given problem,
√ab = 16
⇒ ab = 16²
⇒ ab = 256
Now, \(\frac{b²}{a}\) = 64
⇒ b² = 64a
⇒ a = \(\frac{b²}{64}\)
Substitute, a = \(\frac{b²}{64}\) in ab = 256
⇒ \(\frac{b²}{64}\) × b = 256
⇒ \(\frac{b³}{64}\) = 256
⇒ b³ = 256 × 64
⇒ b³ = 28 × 26
⇒ b³ = 212 × 2²
⇒ b = 24 ×2²
⇒ b = 26
⇒ b = 64
So, from the equation a = \(\frac{b²}{64}\), we get
a = \(\frac{64²}{64}\)
⇒ a = \(\frac{4096}{64}\)
⇒ a = 64
Therefore, the required two numbers are 64 and 64.