All About Ratios: Dividing, Comparing, and Modifying Quantities

You have 12 chocolates and want to share them between Aman and Bella in a 1:3 ratio.

Step-by-Step Method:

1. Add all the parts: 1 + 3 = 4 parts

2. Find one part: 12 chocolates ÷ 4 = 3 chocolates per part

3. Multiply the number of parts each person receives by the value of one part:

• Aman: 1 part = 1 × 3 = 3 chocolates
• Bella: 3 parts = 3 × 3 = 9 chocolates

Example 1:
Twelve sweets are to be divided between A and B in the ratio 1:3. Find how many sweets each receives.

Solution:

Here, A and B get sweets in the ratio 1:3.

This means if all the sweets are divided into 1 + 3 = 4 equal parts,

Then, A gets one part out of the four equal parts made.

                     = `1 / 4` of the total number of sweets

                      = `1 / 4` × 12 sweets = 3 sweets 

          B gets 3 parts out of the 4 equal parts made.

                     = `3 / 4` of the total number of sweets

                     = `3 / 4` × 12 sweets = 9 sweets 

Example 2:
Divide 99 into three parts in the ratio 2:4:5.

Solution:

Since 2 + 4 + 5 = 11

∴ 1st part = `2 / 11` × 99 = 18,

2nd part =  `4 / 11` × 99 = 36

3rd part =  `5 / 11` × 99 = 45

For any two ratios `a / b` and `c / d`, if:

1. a × d = b × c ⇒ `a/b` = `c/d` i.e., both the ratios are equal
   
   
2. .a × d > b × c ⇒ `a/b` > `c/d`  i.e, `a/b` is  greater than `c/d`
   
   
3. a × d < b × c ⇒ `a/b` < `c/d`  i.e, `a/b` is  smaller than `c/d`

i) `5/6` or `7/9`

⇒ 5 × 9 or 7 × 7

⇒ 45 or 49

Since 49 > 45.

⇒`7/9` is greater

ii) `12/17` or `15/19`

⇒ 12 × 19 or 17 × 15

⇒ 228 or 225

Since 255 > 228.

1. If a given quantity is increased in the ratio a:b (where b > a),
   
   The new (resulting) quantity = `b / a` × the given quantity.
   
   
2. If a given quantity is decreased in the ratio a:b (where b < a),
   
   The new (resulting) quantity = `b / a` × the given quantity.

1) Increase 342 in the ratio 3:4.

Solution:
The increased quantity = `4/3` × the given quantity.

                                      = `4/3` × 342

                                      = 456

2) A decrease of 575 in the ratio 5:2.

Solution:
The decreased quantity = `2/5` × the given quantity.

                                       = `2/5` × 575

                                       = 230

Suppose you want to mix red and blue paint in a 2:3 ratio and you have 20 L of paint:

• Total parts: 2 + 3 = 5

• One part: 20 ÷ 5 = 4L

• Red: 2 × 4 =  8 L

• Blue: 3 × 4 =  12 L

• Always add up all parts before dividing anything.

• Use the same units for both quantities.

• Cross-multiply to compare two ratios.

Increase / Decrease in a Ratio

• If a quantity increases or decreases in the ratio a:b, then
  
  New value = `b / a` × Original value

Comparison of Ratios

• (a:b) > (c:d) if ad > bc

• (a:b) = (c:d) if ad = bc

• (a:b) < (c:d) if ad < bc