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How to Divide a Number Into a Ratio

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Last updated date: 19th Jul 2024
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Introduction

The ratio can be defined as a part of a whole, written in a fraction. This can be applied to any quantity or even a single number. This can be useful when one must divide something between two people but not equally.


It is important to remember that a ratio can also be defined as the number of times one quantity can be contained in the whole. Therefore, the divided ratio numbers in the following article will always equal the sum of the whole.


Step-By-Step Guide to Divide Ratios

Considering the whole number to be ‘N’ which must be divided into a ratio of a:b

Let the parts of ‘N’ in the given ratio be x and y.

Step 1:

For $x:y=a:b$

$x+y=N$ must be satisfied.

Once these parameters are met, follow the next steps.


Step 2:

Let $\frac{x}{y}=\frac{a}{b}=k(const)$

∴ $\frac{x}{a}=\frac{y}{b}=k$

$\therefore x=ak$ and $y=bk$


Step 3:

$ak+bk=N$ $\because x+y=N$

$ \therefore k(a+b)=N $

$ \therefore k=\frac{N}{(a+b)} $


Step 4:

Thus, we can say, $x=ak=a(\frac{N}{a+b})=\frac{aN}{a+b}$

Similarly, $y=\frac{bN}{a+b}$

So, the two parts, x and y, are in the ratio a:b

Answer: $a:b=x:y=\frac{aN}{a+b}:\frac{bN}{a+b}$


Solved Examples on Dividing a Number Into a Given Ratio

Q1. Divide 96 into a ratio of 3:5

Solution: Using the formula $a:b=x:y=\frac{aN}{a+b}:\frac{bN}{a+b}$

Here, $a=3$and $b=5$

Using this information and substituting it in the formula, we get

$x:y=\frac{3\times 96}{8}:\frac{5\times 96}{8}$

$\therefore x:y=3\times 12:5\times 12$

Ans: $x:y=36:60$

Therefore, the two parts corresponding to the given ratio that 96 is divided into are 36 and 60.

Q2. Divide 225 in the ratio 7:8

Solution: Using the formula $a:b=x:y=\frac{aN}{a+b}:\frac{bN}{a+b}$

Here, $a=7$and $b=8$

Using this information and substituting it in the formula, we get

$x:y=\frac{7\times 225}{15}:\frac{8\times 225}{15}$

$=7\times 15:8\times 15$

$\therefore x:y=105:120$

The two parts corresponding to the given ratio that 225 is divided into are 105 and 120.

Q3. Divide 1280 into a ratio of 2:5:3

Solution: To divide a number into three parts, we still use the same formula with slight modifications.

$a:b:c=2:5:3$

We need to find $x:y:z$, the three corresponding parts of 1000 in the given ratio.

The formula used is $a:b:c=x:y:z=\frac{aN}{(a+b+c)}:\frac{bN}{(a+b+c)}:\frac{cN}{(a+b+c)}$

This formula can be modified in an above-given manner for many variables, depending on the given ratio.


$\therefore x:y:z=\frac{2\times 1280}{10}:\frac{5\times 1280}{10}:\frac{3\times 1280}{10}$

$x:y:z=2\times 128:5\times 128:3\times 128$

$\therefore x:y:z=256:640:384$

Ans. Therefore, the three parts of 1000 in the ratio 2:5:3 are 256, 640 and 384, respectively.


Conclusion

Dividing a number into a ratio is an easy process that should help you in simple and basic mathematics. One should always double-check the solutions and go through the calculations to avoid unnecessary errors.

FAQs on How to Divide a Number Into a Ratio

1.  Why is the ‘x:y’ ratio not simplified?

The ratio of $x:y$ must not be simplified since we are not looking for the ratio. Instead, the value of x and y and the answer we will get will be the same as the given ratio.

2. Is there any way to check whether the answer is correct?

You can check whether the x and y are correct by simplifying the ratio and comparing it with the given ratio. If both are the same, the answer is correct. Another way to check the values of x and y is to add them and see if the sum equals the given number for division.

3. Can the sum of a+b be greater than N?

If the sum of the given ratio is greater than the number that needs to be divided, an outcome is possible. The parts, in that case, would be very small, in decimals or fractions.