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Difference between revisions of "2017 AMC 10A Problems/Problem 4"

(→‎Solution 1)
(combine solutions 1 and 2 as they are essentially the same. also clarify that the number of toys increases by 1; not necessarily "1 toy is placed in the box" since different toys can be put in and taken out.)
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<math>\textbf{(A)}\ 13.5\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 14.5\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 15.5</math>
 
<math>\textbf{(A)}\ 13.5\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 14.5\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 15.5</math>
  
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==Solution 1==
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==Solution==
  
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Every <math>30</math> seconds, <math>3-2=1</math> toys are put in the box, so after <math>27\cdot30</math> seconds, there will be <math>27</math> toys in the box.  Mia's mom will then put <math>3</math> toys into the box, and we have our total amount of time to be <math>27\cdot30+30=840</math> seconds, which equals <math>14</math> minutes. <math>\boxed{(\textbf{B})\ 14}</math>
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Every <math>30</math> seconds, <math>3</math> toys are put in the box and <math>2</math> toys are taken out, so the number of toys increases by <math>3-2=1</math> every 30 seconds. Then after <math>27 \times 30 = 810</math> seconds (or <math>13 \frac{1}{2}</math> minutes), there are 27 toys in the box. Mia's mom will then put <math>3</math> toys into the box after 30 more seconds, so the total time taken is <math>27\cdot30+30=840</math> seconds, or <math>\boxed{(\textbf{B})\ 14}</math> minutes.
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How did you choose to do <math>27\times30?</math>
 
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~A question by greenstar
 
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==Solution 2==
 
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Though Mia's mom places <math>3</math> toys every <math>30</math> seconds, Mia takes out <math>2</math> toys right after. Therefore, after <math>30</math> seconds, the two have collectively placed <math>1</math> toy into the box. Therefore by <math>13.5</math> minutes, the two would have placed <math>27</math> toys into the box. Therefore, at <math>14</math> minutes, the two would have placed <math>30</math> toys into the box. Though Mia may take <math>2</math> toys out right after, the number of toys in the box first reaches <math>30</math> by <math>14</math> minutes. <math>\boxed{(\textbf{B})\ 14}</math>
 
  
 
==Video Solution==
 
==Video Solution==

Revision as of 12:11, 3 February 2021

Problem

Mia is "helping" her mom pick up $30$ toys that are strewn on the floor. Mia’s mom manages to put $3$ toys into the toy box every $30$ seconds, but each time immediately after those $30$ seconds have elapsed, Mia takes $2$ toys out of the box. How much time, in minutes, will it take Mia and her mom to put all $30$ toys into the box for the first time?

$\textbf{(A)}\ 13.5\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 14.5\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 15.5$

Solution

Every $30$ seconds, $3$ toys are put in the box and $2$ toys are taken out, so the number of toys increases by $3-2=1$ every 30 seconds. Then after $27 \times 30 = 810$ seconds (or $13 \frac{1}{2}$ minutes), there are 27 toys in the box. Mia's mom will then put $3$ toys into the box after 30 more seconds, so the total time taken is $27\cdot30+30=840$ seconds, or $\boxed{(\textbf{B})\ 14}$ minutes.

Video Solution

https://youtu.be/str7kmcRMY8

https://youtu.be/1F0IB0y6578

~savannahsolver

See also

2017 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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