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Chapter 11: Problem 5

A merchant sells three different sizes of canned tomatoes. A large can costs the same as 5 medium cans or 7 small cans. If a customer purchases an equal number of small and large cans of tomatoes for the same amount of money needed to buy 200 medium cans, how many small cans does she purchase? a. 35 b. 45 c. 72 d. 99 e. 208

Short Answer

Expert verified
The customer purchased 35 small cans of tomatoes.

Step by step solution

01

Establish a relationship between can sizes

Let's denote the cost of a medium can as \(m\), of a large can as \(l\), and of a small can as \(s\). According to the information provided, a large can costs the same as 5 medium cans or 7 small cans. Therefore we have the following relations: \(l = 5m\) and \(l = 7s\).
02

Express the cost of a small can in terms of a medium can

As the large can price equals both 5 medium cans and 7 small cans, we can equate the two expressions: \(5m = 7s\). This allows us to express the cost of a small can in terms of a medium can: \(s = 5m/7\)
03

Establish a cost equivalence

The customer purchases equal numbers of small and large cans for the same amount of money needed to buy 200 medium cans. This means: \(l*n+m*n=200m\) where n is the number of cans bought of each type.
04

Express large and small cans as multiples of medium cans

Replace \(l\) and \(s\) with respective equivalences we found earlier. \(5m*n + 5m/7*n = 200m\)
05

Simplify the equation

Simplify the equation by cancelling m from both sides. Then simplify the equation further: \(5n+5n/7 = 200\). After simplifying, we get \((35n+5n)/7 = 200\) and ultimately \(40n/7 = 200\)
06

Find the number of cans

Solve for \(n\). Multiply both sides of the equation by \(7/40\) to isolate \(n\). The solution is \(n = 200 * 7 / 40 = 35\)

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