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Solving a Punch Mixture Problem Using a System of Equations
Apply the seven-step problem-solving strategy for systems of linear equations to a real-world mixture problem solved by graphing.
Problem: Sondra is making 10 quarts of punch from fruit juice and club soda. The number of quarts of fruit juice is 4 times the number of quarts of club soda. How many quarts of fruit juice and how many quarts of club soda does Sondra need?
- Read the problem.
- Identify what to find: the number of quarts of fruit juice and the number of quarts of club soda.
- Name the unknowns: Let = number of quarts of fruit juice. Let = number of quarts of club soda.
- Translate into a system of equations. The total amount of punch is 10 quarts: . The fruit juice is 4 times the club soda: . The system is:
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Solve by graphing both equations on the same coordinate system. The line has slope and y-intercept . The line can be rewritten as , with slope and y-intercept . The two lines intersect at the point .
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Check: Is 8 quarts of fruit juice 4 times 2 quarts of club soda? ✓. Do 8 quarts of fruit juice plus 2 quarts of club soda equal 10 quarts of punch? ✓.
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Answer: Sondra needs 8 quarts of fruit juice and 2 quarts of club soda.
This example demonstrates how a real-world mixture scenario naturally produces two equations — one from the total-quantity constraint and one from a multiplicative relationship between the two ingredients — which together form a system that can be solved by graphing.
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Solving a Punch Mixture Problem Using a System of Equations
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Learn After
A catering lead is preparing 10 quarts of a signature punch for a corporate event. The recipe specifies that the amount of fruit juice (f) must be four times the amount of club soda (c). Which system of equations correctly translates these conditions into a mathematical model?
A catering manager is using a system of equations to determine the ratio of ingredients for a large batch of punch. According to the standard problem-solving strategy, in what order should the following steps be performed to solve this mixture problem by graphing?
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A catering coordinator is using a system of equations to prepare a 10-quart batch of punch for an event. Based on the standard recipe where the amount of fruit juice (f) must be four times the amount of club soda (c), match each mathematical component to the real-world requirement or value it represents.
Mixture Problem Step Identification
A catering coordinator is preparing a 10-quart batch of signature punch where the number of quarts of fruit juice () must be exactly four times the number of quarts of club soda (). When solving this mixture problem graphically, the intersection point of the two lines is (2, 8). In this solution, the value 2 represents the required number of quarts of ____.
Standardizing the 10-Quart Punch Recipe
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