A tailor used 30 buttons that had an average (Arithmetic mean)

A tailor used $30$ buttons that had an average (Arithmetic mean) weight of $x$ grams per button and $20$ other buttons that had an average weight of $80$ grams per button. Which Of the following is the average weight per button, in grams, of the $50$ buttons that the tailor used?

  1. ${x+(20)(80)}/50$
  2. ${x+80}/50$
  3. $3/5x+8/5$
  4. $3/5x+32$
  5. $5/8x$

So, you were trying to be a good test taker and practice for the GRE with PowerPrep online. Buuuut then you had some questions about the quant section—specifically question 20 of the second Quantitative section of Practice Test 1. Those questions testing our knowledge of Numerical Methods for Describing Data can be kind of tricky, but never fear, PrepScholar has got your back!

Survey the Question

Let’s search the problem for clues as to what it will be testing, as this will help shift our minds to think about what type of math knowledge we’ll use to solve this question. Pay attention to any words that sound math-specific and anything special about what the numbers look like, and mark them on our paper.

The question asks us to find the average weight of a set of buttons, which is a good indication that we’ll draw upon our knowledge of Numerical Methods for Describing Data GRE math skill. Let’s keep what we’ve learned about this skill at the tip of our minds as we approach this question.

What Do We Know?

Let’s carefully read through the question and make a list of the things that we know.

 

  1. We have $30$ buttons with an average weight of $x$ grams per button
  2. We have $20$ other buttons with an average weight of $80$ grams per button
  3. We want to find the average weight of all $50$ buttons

 

Develop a Plan

We want to find the average weight of $50$ buttons. We know that to find an average, we take the sum of a set of values and divide by the number of values. So for our question, we have:

$$\Average \Weight \of 50 \Buttons = {\Sum \of \the \Weights \of 50 \Buttons}/50$$

So if we can find the sum of the weights of all of the buttons, we can calculate the average button’s weight. We know that the total sum of all of the weights of the buttons will be the sum of the weights for the $30$ buttons and the $20$ buttons.

$$\Sum \of \the \Weights \of 50 \Buttons = \Sum \of \the \Weights \of 20 \Buttons + \Sum \of \the \Weights \of 30 \Buttons$$

We know that the average weight of the $20$ buttons is $80$ grams per button. So we can find the sum of the weights of the $20$ buttons by multiplying the average weight of them by the number of buttons:

$$\Sum \of \the \Weights \of 20 \Buttons = 20·80$$

Similarly, we know that the sum of the weights of the $30$ buttons will be the product of their average weight, which is $x$, and number of buttons:

$$\Sum \of \the \Weights \of 30 \Buttons = 30·x$$

Let’s combine these two sums of weights together to get the total weight of the buttons, then we can divide by $50$ to get the average weight of the buttons.

Solve the Question

We’ll start with our overall equation for the average weight of the $50$ buttons, then plug in the equations and numbers we developed.

$\Average \Weight \of 50 \Buttons$ $=$ ${\Sum \of \the \Weights \of 50 \Buttons}/50$
$\Average \Weight \of 50 \Buttons$ $=$ ${\Sum \of \the \Weights \of 20 \Buttons + \Sum \of \the \Weights \of 30 \Buttons}/50$
$\Average \Weight \of 50 \Buttons$ $=$ ${20·80+30·x}/50$
$\Average \Weight \of 50 \Buttons$ $=$ ${20·80}/50+{30·x}/50$
$\Average \Weight \of 50 \Buttons$ $=$ $32+3/5x$

Looking at our answer choices, we can see that the correct answer is D, $3/5x+32$.

What Did We Learn

For questions involving an average, we need to keep our minds flexible. There’s a good chance that the question might be more easily solved if we convert the average to a sum. Let’s remember that just because a question is worded a certain way, that doesn’t pigeonhole us into only being able to think of the question that way.

 

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