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A grocery store sells boxes of two kinds of cereals, wheat-based and r
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03 Dec 2018, 05:45
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A grocery store sells boxes of two kinds of cereals, wheat-based and rice-based. In this store, the average weight of a box of rice-based cereal is 60 ounces, the average weight of a box of wheat-based cereal is 48 ounces, and the average weight of all the boxes of cereal is 50 ounces. What is the ratio in this store of boxes of rice-based cereals to wheat-based cereals?
A. \(\frac{1}{12}\) B. \(\frac{1}{6}\) C. \(\frac{1}{5}\) D. \(\frac{1}{4}\) E. \(\frac{1}{3}\)
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03 Dec 2018, 09:38
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aragonn wrote:
A grocery store sells boxes of two kinds of cereals, wheat-based and rice-based. In this store, the average weight of a box of rice-based cereal is 60 ounces, the average weight of a box of wheat-based cereal is 48 ounces, and the average weight of all the boxes of cereal is 50 ounces. What is the ratio in this store of boxes of rice-based cereals to wheat-based cereals?
A. 1/12 B. 1/6 C. 1/5 D. 1/4 E. 1/3
One approach is to use weighted averages Weighted average of groups combined = (group A proportion)(group A average) + (group B proportion)(group B average) + (group C proportion)(group C average) + ...
Let R = number of boxes of rice-based cereal Let W = number of boxes of wheat-based cereal
So, R+W = TOTAL number of boxes of cereal Also, R/(R + W) = proportion of cereal boxes that are rice-based And W/(R + W) = proportion of cereal boxes that are wheat-based
Plug all values into the above formula. . . We get: 50 = [R/(R + W)](60) + [W/(R + W)](48) Multiply both sides by (R+W) to get: 50(R + W) = 60R + 48W Expand to get: 50R + 50W = 60R + 48W Rearrange: 2W = 10R Rearrange more: 2/10 = R/W Simplify: R/W = 1/5
What is the ratio in this store of boxes of rice-based cereals to wheat-based cereals? R/W = 1/5
Re: A grocery store sells boxes of two kinds of cereals, wheat-based and r
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03 Dec 2018, 18:17
Official Explanation:
Method One: Algebraic Approach
This approach is longer, more time-consuming, but easier to understand.
We can set up a weighted average. Let W be the number of boxes of wheat-based cereals, and R be the number of boxes of rice-based cereals. Here’s the weighted average formula:
\(\frac{(60R+48W)}{(R+W)}\)= 50
Multiply both sides by (R + W)
60R + 48W = 50R + 50W
10R + 48W = 50W
10R = 2W
5R = W
\(\frac{R}{W}\) = \(\frac{1}{5}\)
Answer = (C)
Method Two: Intuitive Geometric Approach
This approach is lightning fast but harder to understand.
Look at the the two distance of the individual averages from the weighted average. Wheat-based cereals are at 60, 10 units away from 50. Rice-based cereals are at 48, just 2 units from 50. The ratio of the distances is 10 to 2, or 5:1, so the ratio of the group sizes must be the reciprocal, 1:5.
Answer = (C)
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