A ratio is defined as the comparison of two quantities of the same type through division.  These quantities are then simplified with the help of fractions. Different quantities in a ratio are separated by inserting ‘:’ between them.  For example, 5 litres of a colloidal solution contains 2 litres of oil and 3 litres of water. Therefore, the ratio of oil to water is 2:3.

There are two types of ratios:

• Part-to-Part Ratio: This type of ratio shows the relation between two distinct entities or groups. For example, in a class of 50, there are 30 footballers and 20 cricketers. The ratio of footballers to cricketers is 50:30 = 5:3.
• Part-to-Whole Ratio: This type of ratio shows the relationship between a specific group to a whole. For example, in a cricket team of 11 players, 3 players are all-rounders. Therefore, the ratio of all-rounders to the total number of players is 3:11.

Ratio Formula and Simplification:

Ratio is generally expressed in the form of a:b, which is read as a is to b. Here, entity a is called the antecedent and entity b is known as the consequent. For simplification, the ratio will be converted into the fraction form of ab. Let’s learn about the simplification of ratios step-by-step by solving a simple problem.

Let’s simplify 16:36.

• Step 1: Convert ratio into a fraction. Here, 16:36 becomes 1636
• Step 2: Find the Greatest Common Factor (G.C.F) of 16 and 36. The G.C.F of 16 and 36 is 4.

Note: The largest positive integer that divides a set of whole numbers individually and leaves 0 as the remainder is called the Greatest Common Factor or G.C.F.

• Step 3: Divide the numerator and denominator by the G.C.F. Therefore, after dividing the numerator and the denominator by 4 we get 164364 = 49.
• Step 4: Now, convert the fraction back to the ratio form. Here, 49 becomes 4: 9.

Important Points:

Below are a few important points to remember while solving ratios:

• If both the entities of a ratio a:b are the same, then a:b = 1.
• If a>b, then a:b>1
• If a<b, then a:b<1
• The units of all the entities must be similar before the comparison.

Sample Questions:

• In a certain juice, the ratio (in litres) of water to orange squash to liquid sugar is in the ratio of 5: 4: 3. The juice combination will be altered so that the ratio of water to orange squash is doubled while the ratio of orange squash to liquid sugar is halved. If the altered juice contains 6 litres of liquid sugar, then how many litres of water will it contain?
1. 4
2. 8
3. 10
4. 20
5. 50

Explanation:

Given,
Water: Orange: Sugar = 5: 4: 3

The ratio of water to orange is doubled, while the ratio of orange to sugar is halved.

So,

\frac{Water}{Orange}= \left ( \frac{5}{4} \right )*2= \frac{5}{2} \frac{Orange}{Sugar}= \left ( \frac{4}{3} \right )\ast \frac{1}{2}= \frac{2}{3}

So, the new ratio,

Water: Orange: Sugar = 5: 2: 3

Then,
actual quantity in litres is,

Water: Orange: Sugar = 5x: 2x: 3x

Given,
altered juice contains 6-litres of sugar.

So, 3x = 6 🡪 x = 2

So, water 🡪 5x = 10 litres.

• Alex’s wardrobe has only blue, black, white and yellow shirts. If in the wardrobe the ratio of black shirts to blue shirts is 5: 2, also the ratio of black shirts to white shirts is 3: 8 and the ratio of yellow shirts to white shirts are 1: 4. Then what is the ratio of the number of blue shirts to yellow shirts?
1. 5:16
2. 15:64
3. 3:5
4. 5:3
5. 1:4

Explanation:

Given,

\frac{Black}{Blue}= \frac{5}{2}

\frac{Black}{White}= \frac{3}{8}

<span class="CCcommand">\frac</span><span class="CCbrackets">{</span><span class="CCword">Yellow</span><span class="CCbrackets">}</span><span class="CCbrackets">{</span><span class="CCword">White</span><span class="CCbrackets">}</span><span class="CCother">=</span> <span class="CCcommand">\frac</span><span class="CCbrackets">{</span><span class="CCnumbers">1</span><span class="CCbrackets">}</span><span class="CCbrackets underline">{</span><span class="CCnumbers">4</span><span class="CCbrackets underline">}</span>

Question:

\frac{Blue}{Yellow}=?

We can try to use the given ratios.

<span class="CCcommand">\frac</span><span class="CCbrackets">{</span><span class="CCword">Blue</span><span class="CCbrackets">}</span><span class="CCbrackets">{</span><span class="CCword">Yellow</span><span class="CCbrackets">}</span><span class="CCother">= </span><span class="CCcommand">\frac</span><span class="CCbrackets">{</span><span class="CCword">Blue</span><span class="CCbrackets">}</span><span class="CCbrackets">{</span><span class="CCword">Black</span><span class="CCbrackets">}</span><span class="CCcommand">\ast</span> <span class="CCcommand">\frac</span><span class="CCbrackets">{</span><span class="CCword">Black</span><span class="CCbrackets">}</span><span class="CCbrackets">{</span><span class="CCword">White</span><span class="CCbrackets">}</span><span class="CCcommand">\ast</span> <span class="CCcommand">\frac</span><span class="CCbrackets">{</span><span class="CCword">White</span><span class="CCbrackets">}</span><span class="CCbrackets">{</span><span class="CCword">Yellow</span><span class="CCbrackets">}</span><span class="CCother">=</span> <span class="CCcommand">\frac</span><span class="CCbrackets">{</span><span class="CCnumbers">2</span><span class="CCbrackets">}</span><span class="CCbrackets">{</span><span class="CCnumbers">5</span><span class="CCbrackets">}</span><span class="CCcommand">\ast</span> <span class="CCcommand">\frac</span><span class="CCbrackets">{</span><span class="CCnumbers">3</span><span class="CCbrackets">}</span><span class="CCbrackets">{</span><span class="CCnumbers">8</span><span class="CCbrackets">}</span><span class="CCcommand">\ast</span> <span class="CCcommand">\frac</span><span class="CCbrackets">{</span><span class="CCnumbers">4</span><span class="CCbrackets">}</span><span class="CCbrackets">{</span><span class="CCnumbers">1</span><span class="CCbrackets">}</span><span class="CCother">=</span> <span class="CCcommand">\frac</span><span class="CCbrackets">{</span><span class="CCnumbers">3</span><span class="CCbrackets">}</span><span class="CCbrackets underline">{</span><span class="CCnumbers">5</span><span class="CCbrackets underline">}</span>