In a society, people like three different types of cars: A, B, and C. The ratio of people who like car A to those who like car C is $29 : 42$. The number of people who like only car A and car C together is one-fourth of the number of people who like only car B. The number of people who like only car C is 50 more than four times the number of people who like only car A and car B together. Ten people like all three types of cars, which is equal to the number of people who like only car B and car C together. The number of people who like only car C is twice the number of people who like only car A. The total number of people who like car C is 210. Everyone likes at least one of the cars.

What is the ratio of people who like only car C to those who like car B?

Question

In a society, people like three different types of cars: A, B, and C. The ratio of people who like car A to those who like car C is $29 : 42$. The number of people who like only car A and car C together is one-fourth of the number of people who like only car B. The number of people who like only car C is 50 more than four times the number of people who like only car A and car B together. Ten people like all three types of cars, which is equal to the number of people who like only car B and car C together. The number of people who like only car C is twice the number of people who like only car A. The total number of people who like car C is 210. Everyone likes at least one of the cars.

What is the ratio of people who like only car C to those who like car B?

Accepted Answer

**Solution:**

Let's define the variables for the number of people liking different combinations of cars:

- Let $x$ = number of people who like only car A
- Let $y$ = number of people who like only car B
- Let $z$ = number of people who like only car C
- Let $p$ = number of people who like only cars A & B
- Let $q$ = number of people who like only cars B & C
- Let $r$ = number of people who like only cars A & C
- Let $s$ = number of people who like all three cars A, B & C

From the problem:

1. Ratio of people who like car A to car C is $29 : 42$.
   - Total people who like car A = $x + p + r + s$
   - Total people who like car C = $z + q + r + s = 210$

2. People who like only car A & C together is one-fourth of people who like only car B:
   $$r = \frac{1}{4} y$$

3. People who like only car C is 50 more than four times the people who like only car A & B together:
   $$z = 4p + 50$$

4. Ten people like all three cars, which equals the number of people who like only car B & C together:
   $$s = 10, \quad q = 10$$

5. People who like only car C is twice the people who like only car A:
   $$z = 2x$$

6. Total people who like car C is 210:
   $$z + q + r + s = 210$$

7. Ratio of total people who like car A to car C is $29 : 42$:
   $$\frac{x + p + r + s}{210} = \frac{29}{42}$$

**Step 1: Use $z = 2x$ and $z = 4p + 50$**

From $z = 2x$ and $z = 4p + 50$, we get:
$$2x = 4p + 50 \implies x = 2p + 25$$

**Step 2: Use $r = \frac{1}{4} y$ and $q = s = 10$**

**Step 3: Use total car C equation:**

$$z + q + r + s = 210$$
Substitute $z = 2x$, $q = 10$, $r = \frac{1}{4} y$, $s = 10$:

$$2x + 10 + \frac{1}{4} y + 10 = 210$$
$$2x + \frac{1}{4} y + 20 = 210$$
$$2x + \frac{1}{4} y = 190$$

**Step 4: Use ratio of car A to car C:**

$$\frac{x + p + r + s}{210} = \frac{29}{42}$$
Calculate numerator:

$$x + p + r + s = x + p + \frac{1}{4} y + 10$$

So:

$$\frac{x + p + \frac{1}{4} y + 10}{210} = \frac{29}{42}$$
Multiply both sides by 210:

$$x + p + \frac{1}{4} y + 10 = 210 	imes \frac{29}{42} = 5 	imes 29 = 145$$

So:

$$x + p + \frac{1}{4} y = 135$$

**Step 5: Recall from Step 3:**

$$2x + \frac{1}{4} y = 190$$

**Step 6: Subtract equation from Step 4 from Step 3:**

$$(2x + \frac{1}{4} y) - (x + p + \frac{1}{4} y) = 190 - 135$$

Simplify:

$$2x + \frac{1}{4} y - x - p - \frac{1}{4} y = 55$$

$$x - p = 55$$

**Step 7: Recall from Step 1:**

$$x = 2p + 25$$

Substitute into $x - p = 55$:

$$(2p + 25) - p = 55$$

$$p + 25 = 55$$

$$p = 30$$

Then:

$$x = 2(30) + 25 = 60 + 25 = 85$$

**Step 8: Find $y$ using Step 3 equation:**

$$2x + \frac{1}{4} y = 190$$

Substitute $x=85$:

$$2(85) + \frac{1}{4} y = 190$$

$$170 + \frac{1}{4} y = 190$$

$$\frac{1}{4} y = 20$$

$$y = 80$$

**Step 9: Find $r$ and $z$**

$$r = \frac{1}{4} y = \frac{1}{4} 	imes 80 = 20$$

$$z = 2x = 2 	imes 85 = 170$$

**Step 10: Find total people who like car B:**

People who like car B = $y + p + q + s = 80 + 30 + 10 + 10 = 130$

**Step 11: Find ratio of people who like only car C to people who like car B:**

Only car C = $z = 170$

People who like car B (all types) = $130$

Ratio = $170 : 130 = 17 : 13$

**Answer:** $17 : 13$

**Note:** The options given in the question have (b) as 17:13, but the correct ratio of only car C to people who like car B (all types) is $17 : 13$. However, the question asks for ratio of people who like only car C to people who like car B (all types). So option (b) is correct.

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