Solving the Milk-Water Mixture Ratio Problem

This problem involves calculating the amount of water to add to a milk and water mixture to change its ratio. We start with a total volume and an initial ratio, and we need to find how much water to add to achieve a new ratio.

Calculating Initial Quantities of Milk and Water

First, let's determine the initial amounts of milk and water in the 156-litre mixture. The ratio of milk to water is given as 7 : 6.

• The total number of parts in the initial ratio is the sum of the ratio components: $7 + 6 = 13$ parts.
• The total volume of the mixture is 156 litres. We can find the volume represented by one part by dividing the total volume by the total number of parts: $$ \text{Volume per part} = \frac{\text{Total Volume}}{\text{Total Parts}} = \frac{156 \text{ litres}}{13 \text{ parts}} = 12 \text{ litres/part} $$
• Now, we calculate the initial quantity of milk and water:
  • Initial Milk = $7 \text{ parts} \times 12 \text{ litres/part} = 84 \text{ litres}$
  • Initial Water = $6 \text{ parts} \times 12 \text{ litres/part} = 72 \text{ litres}$

We can represent these initial quantities in a table:

Determining the Amount of Water to Add

The goal is to change the ratio of milk to water to 14 : 13 by adding only water. The amount of milk remains constant.

• Let the amount of water added be '$x$' litres.
• The new quantity of milk will still be 84 litres.
• The new quantity of water will be the initial quantity plus the added water: $72 + x$ litres.
• The desired new ratio is Milk : Water = 14 : 13.
• We can set up an equation using the new quantities and the desired ratio: $$ \frac{\text{New Milk Quantity}}{\text{New Water Quantity}} = \frac{14}{13} $$ $$ \frac{84}{72 + x} = \frac{14}{13} $$

Solving the Ratio Equation

To find the value of '$x$', we solve the equation derived above.

• Cross-multiply the equation: $$ 84 \times 13 = 14 \times (72 + x) $$
• Calculate the product on the left side: $$ 1092 = 14 \times (72 + x) $$
• Distribute the 14 on the right side: $$ 1092 = (14 \times 72) + (14 \times x) $$ $$ 1092 = 1008 + 14x $$
• Isolate the term with '$x$' by subtracting 1008 from both sides: $$ 1092 - 1008 = 14x $$ $$ 84 = 14x $$
• Solve for '$x$' by dividing both sides by 14: $$ x = \frac{84}{14} $$ $$ x = 6 $$

So, 6 litres of water should be added.

Verification of the New Ratio

Let's check if adding 6 litres of water results in the desired 14 : 13 ratio.

• New Milk Quantity = 84 litres.
• New Water Quantity = Initial Water + Added Water = $72 + 6 = 78$ litres.
• The new ratio is Milk : Water = 84 : 78.
• Simplify the ratio by dividing both numbers by their greatest common divisor, which is 6: $$ \frac{84}{6} : \frac{78}{6} = 14 : 13 $$

The new ratio matches the target ratio of 14 : 13. Therefore, the amount of water to be added is indeed 6 litres.