Question

Divide ₹ 35400 into two parts such that if one part is invested in 9%, ₹ 100 shares at 4% discount and the other in 12%, ₹ 50 shares at 8% premium, the annual incomes are equal.

Answer 1

Given:

Total amount = ₹ 35,400.

First part invested in 9% shares of face value ₹ 100 at 4% discount.

Second part invested in 12% shares of face value ₹ 50 at 8% premium.

We use: No. of shares = `"Sum invested"/"Market value of one share"`

Total dividend = Rate × No. of shares × Face value

Step-wise calculation:

1. Market values and dividend per share

First investment:

M.V. = 100 × (1 – 4%)

= ₹ 96 

Dividend per share = 9% of 100

= ₹ 9

Second investment:

M.V. = 50 × (1 + 8%)

= ₹ 54 

Dividend per share = 12% of 50

= ₹ 6

2. Let x = Amount put in the first part (₹).

Then second part = 35,400 – x.

3. Annual incomes

Income from first part = `x/96 xx 9`

= `9/96x`

= `3/32x`

Income from second part = `((35,400 - x)/54) xx 6`

= `(6/54) (35,400 - x)`

= `(1/9) (35,400 - x)`

4. Equate incomes: `(3/32) x = (1/9) (35,400 - x)`

Multiply both sides by 288 (LCM of 32 and 9) or solve directly:

`(3x)/32 = (35,400 - x)/9`

⇒ 3x × 9 = 32(35,400 – x)

⇒ 27x = 32 × 35,400 – 32x 

⇒ 27x + 32x = 32 × 35,400 

⇒ 59x = 1,132,800   ...(Since 32 × 35,400 = 1,132,800) 

⇒ `x = (1,132,800)/59`

⇒ x = ₹ 19,200

5. Second part = 35,400 − 19,200

= ₹ 16,200

6. Verification:

Income 1 = `3/32 xx 19,200` 

= 3 × 600

= ₹ 1,800

Income 2 = `1/9 xx 16,200`

= ₹ 1,800

Equal as required.