Ratio and Proportion

I. The salaries of three friends – P, Q and R are in the ratio 5 : 6 : 8

Q1: If P, Q and R save 10%, 15% and 20% of their salaries respectively, what is the ratio of their savings?

Step 1: Assume salaries are:
P = 5x, Q = 6x, R = 8x
Step 2: Savings:
P saves 10% of 5x = \(0.10 \times 5x = 0.5x\)
Q saves 15% of 6x = \(0.15 \times 6x = 0.9x\)
R saves 20% of 8x = \(0.20 \times 8x = 1.6x\)
Step 3: Ratio of savings = \[ 0.5x : 0.9x : 1.6x = 0.5 : 0.9 : 1.6 \] Multiply all terms by 10: \[ 5 : 9 : 16 \]Answer: d. 5 : 9 : 16

Q2: If P, Q and R save one-half, one-third and one-fourth of their salaries respectively, what is the ratio of their expenditures?

Step 1: Expenditures = Salary − Savings
P saves ½ of 5x ⇒ spends ½ ⇒ expenditure = \( \frac{1}{2} \times 5x = 2.5x \)
Q saves ⅓ of 6x ⇒ spends ⅔ ⇒ expenditure = \( \frac{2}{3} \times 6x = 4x \)
R saves ¼ of 8x ⇒ spends ¾ ⇒ expenditure = \( \frac{3}{4} \times 8x = 6x \)
Step 2: Ratio = \[ 2.5x : 4x : 6x = 2.5 : 4 : 6 \] Multiply all terms by 2: \[ 5 : 8 : 12 \]Answer: d. 5 : 8 : 12

Q3: If P, Q and R receive increments of 20%, 10% and 15% respectively, what will be the ratio of their new salaries?

Step 1: New salaries:
P = \(5x + 20\%\ of\ 5x = 5x(1 + 0.20) = 6x\)
Q = \(6x(1 + 0.10) = 6.6x\)
R = \(8x(1 + 0.15) = 9.2x\)
Step 2: Ratio = \[ 6x : 6.6x : 9.2x = 6 : 6.6 : 9.2 \] Multiply all by 10: \[ 60 : 66 : 92 \] Divide by 2: \[ 30 : 33 : 46 \]Answer: c. 30 : 33 : 46

Q4: If P’s salary decreases by 10%, Q’s increases by 20%, and R’s decreases by 30%, what is the new salary ratio?

P = \(5x \times 0.9 = 4.5x\)
Q = \(6x \times 1.2 = 7.2x\)
R = \(8x \times 0.7 = 5.6x\)
Ratio: \[ 4.5 : 7.2 : 5.6 \] Multiply all by 10: \[ 45 : 72 : 56 \]Answer: a. 45 : 72 : 56