Blend Ratio Calculation: Apple and Orange Juice Mix
This problem requires finding the ratio in which two different juice blends are mixed to achieve a specific final ratio of apple juice to orange juice.
Understanding the Blends
- Blend 1: Contains 20% apple juice, which means it contains 80% orange juice.
- Blend 2: Contains 30% orange juice, which means it contains 70% apple juice.
- Final Mix: The resultant mixture of Blend 1 and Blend 2 has a ratio of apple juice to orange juice of 4:3.
Setting up the Equation for Mixing
Let $x$ represent the quantity of the first blend and $y$ represent the quantity of the second blend used in the final mix. The goal is to determine the ratio $x : y$.
Calculate the contribution of apple and orange juice from each blend:
- From Blend 1 ($x$ units): Apple Juice = $0.20x$, Orange Juice = $0.80x$.
- From Blend 2 ($y$ units): Apple Juice = $0.70y$, Orange Juice = $0.30y$.
In the combined final mix:
- Total Apple Juice = $0.20x + 0.70y$
- Total Orange Juice = $0.80x + 0.30y$
The problem states that the ratio of total apple juice to total orange juice in the final mix is 4:3.
This gives us the equation:
$ \frac{0.20x + 0.70y}{0.80x + 0.30y} = \frac{4}{3} $
Solving for the Blend Ratio
We solve the equation to find the ratio $x : y$:
- Perform cross-multiplication:
$ 3 \times (0.20x + 0.70y) = 4 \times (0.80x + 0.30y) $
- Distribute the multipliers:
$ 0.60x + 2.10y = 3.20x + 1.20y $
- Rearrange the terms to group $x$ terms and $y$ terms:
$ 2.10y - 1.20y = 3.20x - 0.60x $
$ 0.90y = 2.60x $
- Determine the ratio $x : y$:
$ \frac{x}{y} = \frac{0.90}{2.60} $
$ \frac{x}{y} = \frac{9}{26} $
The ratio in which the first blend and the second blend are mixed is 9 : 26.
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