Introduction
- When we compare two quantities or numbers using division, then it is called Ratio and Whenever we compare ratios, then it is called Proportion.
- In the following example we are equating two ratios and it is called Proportion.
Ratios and Proportions
- Ratios allows us to compare two quantities by showing how much of one quantity is contained by the other.
- Proportions on the other hand tells us that the ratios are equal. There can be two or more ratios in proportion to each other.
Solved Examples
Solved Example
Problem: Find the ratio of the shaded portion to unshaded for both the diagrams check whether they are in proportion or not?
SOLUTION
- In the first grid, there are 100 boxes total, with 50 shaded and 50 unshaded. The ratio of shaded to unshaded is:
$$\frac{50}{50} = \frac{1}{1}$$
- In the second grid, there are 36 boxes total, with 18 shaded and 18 unshaded. The ratio of shaded to unshaded is:
$$\frac{18}{18} = \frac{1}{1}$$
- We can clearly see that both ratios are equal, thus they are in proportion, and can be represented by:
$$50 : 50 :: 18 : 18$$
Final Answer: $50 : 50 :: 18 : 18$
Solved Example
Problem: Find out the ratios of blue balls and green balls differently and also check if they are in proportion or not?
SOLUTION
- Row 1: 3 blue balls, 3 green balls. Ratio: $3 : 3$ which simplifies to $1 : 1$.
- Row 2: 1 blue ball, 2 green balls. Ratio: $1 : 2$.
- Row 3: 2 blue balls, 3 green balls. Ratio: $2 : 3$.
- Row 4: 3 blue balls, 2 green balls. Ratio: $3 : 2$.
- Because the ratios ($1:1, 1:2, 2:3, 3:2$) are all different and do not share a common value, they are not in proportion.
Final Answer: The ratios are not in proportion.
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Solved Example
Problem: Suppose in a school there are total 130 students in which there are 50 girls and 80 boys. If one day 25 out of 50 girls and 40 out of 80 boys were present that day then check whether these ratios (present girls and boys to the total) are in proportion or not?
SOLUTION
- Ratio of present girls to total girls: $25 : 50 = \frac{1}{2}$
- Ratio of present boys to total boys: $40 : 80 = \frac{1}{2}$
- Both ratios are equivalent, so they are in proportion:
$$25 : 50 :: 40 : 80$$
Final Answer: $25 : 50 :: 40 : 80$
Solved Example
Problem: If there are four friends A, B, C and D. A and B have a total of £50, they share it so that A got £20 and B got £30. Similarly C and D share the total amount of £100 such that C got £40 and D got £60. Find the ratios of sharing and compare them that they are in proportion or not?
SOLUTION
- Distribution for A and B ($A=20, B=30$): Ratio $20 : 30 = \frac{2}{3}$.
- Distribution for C and D ($C=40, D=60$): Ratio $40 : 60 = \frac{2}{3}$.
Final Answer: The ratios reduce to $\frac{2}{3}$ and are in proportion.
Solved Example
Problem: Olivia is going to make some ice cream. She needs to mix Custard powder, Milk and sugar in the ratio – 1 : 4 : 20. If she has 25g of Custard, 100g of Sugar, and 500g of Milk. Does Olivia have enough Custard, Sugar and Milk to make ice cream?
SOLUTION
- Required Ratio (Custard : Milk : Sugar) = $1 : 4 : 20$.
- Starting with 25g of Custard:
- Milk needed: $25\text{g} \times 4 = 100\text{g}$
- Sugar needed: $25\text{g} \times 20 = 500\text{g}$
- Olivia has 500g of Milk (enough) but only 100g of Sugar (she needs 500g).
Final Answer: No, she does not have enough sugar to maintain the $1:4:20$ proportion.
