Two mixtures P and Q are in the ratio of 3:2 respectively. Mixture P contains *a*% of milk and *b*% of water, and mixture Q contains *d*% of milk and *e*% of water. If mixture Q is mixed with mixture P, then the final quantity of milk becomes 23% of the total mixture. Given that *a + d = 45*, which statement(s) is/are necessary to find the final quantity of milk?

(I) $a - d = 10$

(II) Initial quantity of mixture P is 60 liters out of which 15 liters is milk.

(III) If 15 liters of the mixture is taken out from mixture P and mixed with mixture Q, then the total quantity of water in mixture Q becomes 40 liters.

Question

Two mixtures P and Q are in the ratio of 3:2 respectively. Mixture P contains *a*% of milk and *b*% of water, and mixture Q contains *d*% of milk and *e*% of water. If mixture Q is mixed with mixture P, then the final quantity of milk becomes 23% of the total mixture. Given that *a + d = 45*, which statement(s) is/are necessary to find the final quantity of milk?

(I) $a - d = 10$

(II) Initial quantity of mixture P is 60 liters out of which 15 liters is milk.

(III) If 15 liters of the mixture is taken out from mixture P and mixed with mixture Q, then the total quantity of water in mixture Q becomes 40 liters.

Accepted Answer

**Solution:**

1. Let the quantities of mixtures P and Q be $3z$ and $2z$ liters respectively.

2. Given that the final milk percentage after mixing is 23%, we have:
   $$\frac{3z 	imes a/100 + 2z 	imes d/100}{3z + 2z} = \frac{23}{100}$$
   Simplifying:
   $$\frac{3za + 2zd}{5z} = 23 \Rightarrow 3a + 2d = 115 \quad (i)$$

3. Given $a + d = 45 \quad (ii)$

4. From (i) and (ii), we have two equations with two variables:
   - From (ii): $d = 45 - a$
   - Substitute in (i): $3a + 2(45 - a) = 115 \Rightarrow 3a + 90 - 2a = 115 \Rightarrow a = 25$, and thus $d = 20$

5. However, without knowing $z$, the total quantity, we cannot find the final quantity of milk.

6. Statement (I) $a - d = 10$ contradicts the values found from (i) and (ii), so it is inconsistent.

7. Statement (II) gives the initial quantity of mixture P as 60 liters with 15 liters milk, which allows us to find $z$:
   $$3z = 60 \Rightarrow z = 20$$

8. Knowing $z$ allows calculation of final quantities and milk content.

9. Statement (III) involves removing 15 liters from P and adding to Q, affecting water quantity, but does not help find $z$ or final milk quantity directly.

**Conclusion:** Only statement (II) is necessary to find the final quantity of milk.

**Answer:** Only (II)

Answer

Answer

Answer