Read the information carefully and answer the question below.

There are three companies X, Y, and Z. The total number of type A pencils manufactured is 40\% less than the total number of type B pencils manufactured by all three companies combined. The ratio of the total number of type B pencils manufactured by Z to the total number of type B pencils manufactured by all three companies is 3 : 10. The number of type A pencils manufactured by X is equal to the total number of pencils manufactured by Y. The number of type A pencils manufactured by Z and Y are equal. The total number of pencils manufactured by X is half of the total number of pencils manufactured by all three companies together.

(Note: Total pencils manufactured by any company = type A + type B pencils manufactured)

What is the ratio of type B pencils manufactured by Y to the total pencils manufactured by Z?

Question

Read the information carefully and answer the question below.

There are three companies X, Y, and Z. The total number of type A pencils manufactured is 40\% less than the total number of type B pencils manufactured by all three companies combined. The ratio of the total number of type B pencils manufactured by Z to the total number of type B pencils manufactured by all three companies is 3 : 10. The number of type A pencils manufactured by X is equal to the total number of pencils manufactured by Y. The number of type A pencils manufactured by Z and Y are equal. The total number of pencils manufactured by X is half of the total number of pencils manufactured by all three companies together.

(Note: Total pencils manufactured by any company = type A + type B pencils manufactured)

What is the ratio of type B pencils manufactured by Y to the total pencils manufactured by Z?

Accepted Answer

**Solution:**

Let's denote:

- Total pencils manufactured by all three companies = $T$
- Total type B pencils by all companies = $B$
- Total type A pencils by all companies = $A$

Given that $A$ is 40\% less than $B$, so:

$$A = B - 0.4B = 0.6B$$

Since total pencils $T = A + B = 0.6B + B = 1.6B$

Also, the ratio of type B pencils by Z to total type B pencils is:

$$\frac{B_Z}{B} = \frac{3}{10} \implies B_Z = \frac{3}{10}B$$

Total pencils by X is half of total pencils:

$$T_X = \frac{T}{2} = \frac{1.6B}{2} = 0.8B$$

Number of type A pencils by X equals total pencils by Y:

$$A_X = T_Y$$

Number of type A pencils by Z equals number of type A pencils by Y:

$$A_Z = A_Y$$

Let’s denote:

- Type A pencils by X, Y, Z as $A_X, A_Y, A_Z$
- Type B pencils by X, Y, Z as $B_X, B_Y, B_Z$

We know:

1. $A_X + A_Y + A_Z = A = 0.6B$
2. $B_X + B_Y + B_Z = B$
3. $B_Z = \frac{3}{10}B$
4. $T_X = A_X + B_X = 0.8B$
5. $T_Y = A_Y + B_Y$
6. $A_X = T_Y$
7. $A_Y = A_Z$

From (6), $A_X = T_Y = A_Y + B_Y$

From (7), $A_Y = A_Z$, so $A_Y + A_Z = 2A_Y$

From (1):

$$A_X + A_Y + A_Z = A_X + 2A_Y = 0.6B$$

Substitute $A_X = A_Y + B_Y$ from (6):

$$(A_Y + B_Y) + 2A_Y = 0.6B \ 3A_Y + B_Y = 0.6B 	ag{8}$$

From (4), $T_X = A_X + B_X = 0.8B$, and $A_X = A_Y + B_Y$:

$$ (A_Y + B_Y) + B_X = 0.8B \ B_X = 0.8B - A_Y - B_Y 	ag{9}$$

From (2), sum of all type B pencils:

$$B_X + B_Y + B_Z = B \ B_X + B_Y + \frac{3}{10}B = B \ B_X + B_Y = B - \frac{3}{10}B = \frac{7}{10}B 	ag{10}$$

Substitute $B_X$ from (9) into (10):

$$0.8B - A_Y - B_Y + B_Y = \frac{7}{10}B \ 0.8B - A_Y = 0.7B \ -A_Y = 0.7B - 0.8B = -0.1B \ A_Y = 0.1B$$

Using $A_Y = 0.1B$ in (8):

$$3(0.1B) + B_Y = 0.6B \ 0.3B + B_Y = 0.6B \ B_Y = 0.3B$$

Now, find total pencils manufactured by Z:

$$T_Z = A_Z + B_Z = A_Y + B_Z = 0.1B + 0.3B = 0.4B$$

We need the ratio of type B pencils manufactured by Y to total pencils manufactured by Z:

$$\frac{B_Y}{T_Z} = \frac{0.3B}{0.4B} = \frac{3}{4}$$

Check options: 3:4 corresponds to option (a), but the correct answer is option (b) 1:9 according to the original marking. Let's re-examine the last step carefully.

Wait, the question asks for the ratio of type B pencils manufactured by Y to total pencils manufactured by Z.

We found:

- $B_Y = 0.3B$
- $T_Z = A_Z + B_Z = A_Y + B_Z = 0.1B + 0.3B = 0.4B$

Ratio:

$$\frac{B_Y}{T_Z} = \frac{0.3B}{0.4B} = \frac{3}{4}$$

So the ratio is 3:4, which is option (a).

**Answer:** 3 : 4

**Note:** The original marking of option (b) as correct seems inconsistent with the data and calculations. The correct ratio based on the given data is 3:4.

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