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Proportion Problems

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How to Solve Proportion Problems?

Have You Ever Wondered What Proportions Are?
Imagine you have a pizza cut into 8 pieces; of these 8 pieces, you can only eat 2 before you are full. Now, the next day you eat another pizza cut into 4 pieces, wherein you can eat only 1 of the pieces before your stomach is full. How do you determine if you ate the same amount on both days? Here is where proportion comes in. It is used when you have to compare two fractions.


Pizza sliced in 8 and 4 pieces to show the proportion


Pizza sliced in 8 and 4 pieces to show the proportion


How Are Proportions Written?

Proportions can be written in two ways:

  1. $\frac{a}{b}:\frac{c}{d}$

  2. $a:b::c:d$

In both these instances, ‘a’ and ‘d’ are the extremes while ‘b’ and ‘c’ are known as the middle terms or the means.


How to Solve Proportion Problems?

There is one simple rule to check if the two fractions are in proportion,

Product of the middle terms = Product of the Extremes

$b\times c=a\times d$

If $b\times c\ne a\times d$, the two given fractions are not in proportion.


Examples of Proportion Problems

Q1. Determine which of these is in proportion. Fill in the blank with either an equal to sign $=$ or a not equal sign $\ne $.


  1. $\frac{8}{18}$ $\frac{1}{9}$


  1. $\frac{4}{12}$ $\frac{9}{27}$


Solutions:


  1. $\frac{8}{18}$ $\frac{1}{9}$

This can be written as $8:18::1:9$which is in the format $a:b::c:d$

On solving, we get,

Product of means = $18\times 1=18$

Product of extremes = $8\times 9=72$

Since $18\ne, 9$the two ratios are not in proportion.

Ans: $\frac{8}{18}\ne \frac{1}{9}$


  1. $\frac{4}{12}$ $\frac{9}{27}$

This can be written as $4:12::9:27$which is in the format $a:b::c:d$

On solving, we get,

Product of means = $12\times 9=108$

Product of extremes = $4\times 27=108$

Since $108=108$the two ratios are in proportion.

Ans: $\frac{4}{12}=\frac{9}{27}$


Q2. Complete the following proportions


  1. $1:3::4:\_$

  2. $5:2::\_:10$

Solutions:

  1. Since $1:3::4:\_$is already in the format of $a:b::c:d$we can directly equate the products and solve for ‘d’

$b\times c=a\times d$

$3\times 4=1\times d$

∴$d=12$

Ans: 12


  1. Since $5:2::\_:10$is already in the required format, we must now solve for ‘c’

$2\times c=5\times 10$

$c=\frac{50}{2}$

∴$c=25$

Ans: 25


Q3. One boy eats 12 out of 15 slices of pizza. How many slices of pizza should another boy eat if he has a pizza with 5 slices and wants to eat the same amount as the first boy?

Solution:

To solve a word problem, first, you must take the given data and make it into ratios.

The first boy ate 12 out of 15, so we have $\frac{12}{15}$

The second boy has a total of 5 slices, so we must find $\frac{c}{5}$

This gives us the ratios in the following format $\frac{a}{b}:\frac{c}{d}$

We know, $b\times c=a\times d$

$15\times c=12\times 5$

∴$c=\frac{12\times 5}{15}$

∴$c=4$

Ans: The second boy should eat 4 slices of pizza.

Q4. If $a:b=3:7$and $b:c=5:9$then, find $a:c$

Solution:

We know that ratios can be written as fractions

So $a:b=\frac{a}{b}$ and $b:c=\frac{b}{c}$

∴ $\frac{a}{b}\times \frac{b}{c}=\frac{a}{c}$

∴$\frac{3}{7}\times \frac{5}{9}$ = $\frac{5}{21}$

Ans: $a:c=5:21$


Q5. If $a:b=3:7$and $b:c=5:9$then, find $a:b:c$

Solution:

To find this ratio, the value of ‘b’ must be the same in both ratios. The proportion remains the same when a fraction is multiplied by the same number in the numerator and denominator.

To make the value of ‘b’ equal in both these fractions, we must multiply the first by 5 and the second by 7.

$\frac{3\times 5}{7\times 5}$ and $\frac{5\times 7}{9\times 7}$


= $\frac{15}{35}$ and $\frac{35}{63}$making the value of ‘b’ equal


We can therefore write the equation $a:b:c=15:35:63$

FAQs on Proportion Problems

1. What exactly is a proportion in Maths?

A proportion is a statement that says two ratios or fractions are equal. For example, the ratio 1/2 is proportional to 4/8 because they both represent the same value. It's essentially an equation that shows the equivalence between two ratios.

2. What are the main types of proportions students learn about?

There are two primary types of proportions that are fundamental to understanding this concept:

  • Direct Proportion: When one quantity increases, the other quantity also increases at the same rate, and vice versa. For example, the more hours you work, the more money you earn.
  • Inverse Proportion: When one quantity increases, the other quantity decreases. For example, the faster you drive, the less time it takes to reach your destination.

3. What is the main difference between a ratio and a proportion?

The main difference is that a ratio is a comparison of two quantities (like 3 apples to 4 oranges), while a proportion is an equation that states two ratios are equal (like 3/4 = 6/8). Think of a ratio as an expression and a proportion as a complete mathematical sentence.

4. How do you solve a basic proportion problem with one unknown value?

The most common method is cross-multiplication. If you have an equation like a/b = c/x, you multiply the numbers diagonally across the equals sign. This gives you a × x = b × c. Then, you can solve for 'x'. This method works for both direct and inverse proportion problems after setting them up correctly.

5. Can you give a simple real-world example of a proportion problem?

Certainly. Imagine a recipe for juice that requires 2 cups of sugar for every 5 lemons. If you only have 10 lemons, how much sugar do you need? You can set up a proportion: 2 sugar / 5 lemons = x sugar / 10 lemons. By solving this, you find you need 4 cups of sugar. This is a direct proportion in action.

6. Why is it important to identify if a word problem involves direct or inverse proportion?

Identifying the type of proportion is crucial because it determines how you set up your equation. If you mistakenly treat an inverse proportion problem (like speed and time) as a direct one, your answer will be incorrect. Understanding the relationship between the quantities helps you build the correct mathematical model to find the right solution.

7. Are there any common mistakes to avoid when solving proportion problems?

A common mistake is mixing up the units or the order of the items in the ratios. Always make sure you are comparing the same things in the same order on both sides of the equation. For example, if you put 'kilograms' in the numerator of the first ratio, 'kilograms' must also be in the numerator of the second ratio.