Three companies A, B, and C sold cars and bikes. The ratio of cars sold by A, B, and C is 4:5:6, and the ratio of bikes sold by A, B, and C is 3:2:5. The difference between the number of cars and bikes sold by B is 18. Also, the ratio of cars sold by B to bikes sold by C is 8:11.

If the ratio of total bikes sold by A to company D is 3:7, and the total cars sold by D are 20\% more than those sold by B, find the sum of the total bikes and cars sold by company D.

Question

Three companies A, B, and C sold cars and bikes. The ratio of cars sold by A, B, and C is 4:5:6, and the ratio of bikes sold by A, B, and C is 3:2:5. The difference between the number of cars and bikes sold by B is 18. Also, the ratio of cars sold by B to bikes sold by C is 8:11.

If the ratio of total bikes sold by A to company D is 3:7, and the total cars sold by D are 20\% more than those sold by B, find the sum of the total bikes and cars sold by company D.

Accepted Answer

**Solution:**

1. Let the common multiplier for cars sold by A, B, and C be *x*.
   - Cars sold by A = 4x
   - Cars sold by B = 5x
   - Cars sold by C = 6x

2. Let the common multiplier for bikes sold by A, B, and C be *y*.
   - Bikes sold by A = 3y
   - Bikes sold by B = 2y
   - Bikes sold by C = 5y

3. Given the difference between cars and bikes sold by B is 18:
   $$5x - 2y = 18 	ag{1}$$

4. Given the ratio of cars sold by B to bikes sold by C is 8:11:
   $$\frac{5x}{5y} = \frac{8}{11}$$ $$\implies $$ $$\frac{x}{y} = \frac{8}{11}$$

5. From equation (2), express *x* in terms of *y*:
   $$x = \frac{8}{11} y$$

6. Substitute *x* in equation (1):
   $$5 	imes \frac{8}{11} y - 2y = 18$$
   $$\frac{40}{11} y - 2y = 18$$
   $$\frac{40}{11} y - \frac{22}{11} y = 18$$
   $$\frac{18}{11} y = 18$$
   $$y = \frac{18 	imes 11}{18} = 11$$

7. Calculate *x*:
   $$x = \frac{8}{11} 	imes 11 = 8$$

8. Calculate cars and bikes sold by A, B, and C:
   - Cars by A = 4x = $$4 	imes 8$$ = 32
   - Cars by B = 5x = $$5 	imes 8$$ = 40
   - Cars by C = 6x = $$6 	imes 8$$ = 48
   - Bikes by A = 3y = $$3 	imes 11$$ = 33
   - Bikes by B = 2y = $$2 	imes 11$$ = 22
   - Bikes by C = 5y = $$5 	imes 11$$ = 55

9. Given the ratio of bikes sold by A to D is 3:7:
   $$\frac{33}{	ext{Bikes by D}} = \frac{3}{7}$$
   $$	ext{Bikes by D} = \frac{33 	imes 7}{3} = 77$$

10. Total cars sold by D are 20\% more than those sold by B:
    $$	ext{Cars by D} = 40 + 0.20 	imes 40 = 40 + 8 = 48$$

11. Sum of total bikes and cars sold by D:
    $$77 + 48 = 125$$

**Answer:** 125

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