One cup has juice and water in the ratio 5 ∶ 2, while another cup of the same capacity has them in the ratio 7 ∶ 4, respectively. If contents of both the cups (when full) are poured in a vessel, then what will be the final ratio of water to juice in the vessel?
25 ∶ 52
Understanding the Ratio and Mixing Problem
This question involves mixing the contents of two containers, each holding a mixture of juice and water in different ratios. The key is to determine the actual amounts of juice and water contributed by each container when they are combined, and then find the ratio of water to juice in the final mixture.
Calculating Quantities in Each Cup
We are given that both cups have the same capacity. To simplify calculations, let's assume a convenient total capacity for each cup. The ratios are 5 ∶ 2 (total parts 7) and 7 ∶ 4 (total parts 11). A good common capacity would be the Least Common Multiple (LCM) of the total parts, which are 7 and 11.
LCM(7, 11) = 77
Let's assume the capacity of each cup is 77 units.
Cup 1: Juice ∶ Water = 5 ∶ 2
Total parts in Cup 1 = 5 + 2 = 7 parts.
Value of 1 part = $\frac{\text{Total Capacity}}{\text{Total Parts}} = \frac{77}{7} = 11 \text{ units}$.
- Amount of Juice in Cup 1 = 5 parts $\times$ 11 units/part = $5 \times 11 = 55 \text{ units}$.
- Amount of Water in Cup 1 = 2 parts $\times$ 11 units/part = $2 \times 11 = 22 \text{ units}$.
Check: Total in Cup 1 = 55 + 22 = 77 units (Matches assumed capacity).
Cup 2: Juice ∶ Water = 7 ∶ 4
Total parts in Cup 2 = 7 + 4 = 11 parts.
Value of 1 part = $\frac{\text{Total Capacity}}{\text{Total Parts}} = \frac{77}{11} = 7 \text{ units}$.
- Amount of Juice in Cup 2 = 7 parts $\times$ 7 units/part = $7 \times 7 = 49 \text{ units}$.
- Amount of Water in Cup 2 = 4 parts $\times$ 7 units/part = $4 \times 7 = 28 \text{ units}$.
Check: Total in Cup 2 = 49 + 28 = 77 units (Matches assumed capacity).
Mixing the Contents
Now, the contents of both full cups are poured into a vessel. To find the total amount of juice and water in the vessel, we sum the amounts from each cup.
- Total amount of Juice in the vessel = Juice from Cup 1 + Juice from Cup 2
- Total amount of Juice = 55 units + 49 units = 104 units.
- Total amount of Water in the vessel = Water from Cup 1 + Water from Cup 2
- Total amount of Water = 22 units + 28 units = 50 units.
Determining the Final Ratio of Water to Juice
The question asks for the final ratio of water to juice. This is the total amount of water divided by the total amount of juice.
Final Ratio (Water ∶ Juice) = Total Water ∶ Total Juice
Final Ratio = 50 ∶ 104
Simplifying the Ratio
The ratio 50 ∶ 104 can be simplified by dividing both numbers by their greatest common divisor. Both 50 and 104 are divisible by 2.
- $50 \div 2 = 25$
- $104 \div 2 = 52$
The simplified ratio of water to juice is 25 ∶ 52.
Summary of Calculations
| Ratio (Juice ∶ Water) | Total Parts | Juice (units) | Water (units) | |
|---|---|---|---|---|
| Cup 1 | 5 ∶ 2 | 7 | 55 | 22 |
| Cup 2 | 7 ∶ 4 | 11 | 49 | 28 |
| Total in Vessel | 104 | 50 |
Final Ratio (Water ∶ Juice) = 50 ∶ 104 = 25 ∶ 52.
Revision Table: Ratio and Mixing Concepts
| Concept | Explanation | How it Applies Here |
|---|---|---|
| Ratio | A comparison of two quantities. e.g., a:b means a units of the first quantity for every b units of the second. | Used to represent the composition of juice and water in each cup. |
| Total Parts in a Ratio | The sum of the numbers in the ratio (a+b for a:b). Represents the whole quantity relative to the ratio parts. | Used to determine the fraction of juice and water in each cup. |
| Least Common Multiple (LCM) | The smallest positive integer that is a multiple of two or more integers. | Used to find a convenient common capacity for the cups, simplifying calculations without fractions initially. |
| Mixing Ratios | Combining quantities from different mixtures. Requires calculating the absolute amount of each component from all sources and summing them. | We summed the total juice and total water from both cups. |
| Simplifying Ratios | Dividing both parts of a ratio by their greatest common divisor to express it in its simplest form. | The final 50:104 ratio was simplified to 25:52. |
Additional Information: Working with Ratios
Understanding ratios is fundamental in many mathematical problems. When dealing with mixtures, ratios tell us the proportion of each component. Here are some additional points:
- Representing Ratios: Ratios can be written as a ∶ b, a/b, or "a to b".
- Proportions: A proportion is an equation stating that two ratios are equal, e.g., a ∶ b = c ∶ d or a/b = c/d. This is useful when scaling quantities.
- Total Quantity from Ratio: If a ratio is a ∶ b and the total quantity is T, the amount of the first component is $\frac{a}{a+b} \times T$, and the second is $\frac{b}{a+b} \times T$. We used this idea when distributing the 77 units capacity based on the ratios.
- Order Matters: The ratio of A to B (A ∶ B) is different from the ratio of B to A (B ∶ A), unless A and B are equal. This is crucial in this problem, as the final answer requires the ratio of *water* to *juice*, not the other way around.
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