Difference between revisions of "2013 AMC 8 Problems/Problem 4"
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==Problem== | ==Problem== | ||
| + | Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra \$2.50 to cover her portion of the total bill. What was the total bill? | ||
| − | ==Solution== | + | <math> \textbf{(A)}\ \text{\textdollar}120\qquad\textbf{(B)}\ \text{\textdollar}128\qquad\textbf{(C)}\ \text{\textdollar}140\qquad\textbf{(D)}\ \text{\textdollar}144\qquad\textbf{(E)}\ \text{\textdollar}160 </math> |
| + | |||
| + | ==Solution 1== | ||
| + | Since Judi's 7 friends had to pay \$2.50 extra each to cover the total amount that Judi should have paid, we multiply <math>2.50\cdot7=17.50</math> is the bill Judi would have paid if she had money. Hence, to calculate the total amount, we multiply <math>17.50\cdot8=\boxed{\textbf{(C) } 140}</math> to find the total the 8 friends paid. | ||
| + | |||
| + | ==Solution 2== | ||
| + | Let <math>m</math> be the total bill the 8 friends split up; each person was supposed to pay <math>\$\frac{m}{8}</math>. Since Judi forgot her money, we have to add <math>\$\frac{m}{8}+2.50</math> to get the amount of money each of the <math>7</math> friends have to pay. After multipling by 7, we have <math>7(\frac{m}{8}+2.50)=\frac{7m}{8} + 17.5</math>. This is equal to <math>m</math> because both sides represent the total bill. Solving for <math>m</math>:<cmath>\frac{7m}{8} + 17.5=m</cmath> | ||
| + | <cmath>\frac{m}{8}=17.5</cmath> | ||
| + | <cmath>m=\boxed{\textbf{(C) } 140}.</cmath> | ||
| + | |||
| + | ~BananaBall00 ~joyoforigami (TOGFA mathematics) edits by ~megaboy6679 | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/lvaWrFwW6BM ~savannahsolver | ||
==See Also== | ==See Also== | ||
| − | {{AMC8 box|year=2013| | + | {{AMC8 box|year=2013|num-b=3|num-a=5}} |
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 22:57, 1 February 2023
Problem
Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra $2.50 to cover her portion of the total bill. What was the total bill?
Solution 1
Since Judi's 7 friends had to pay $2.50 extra each to cover the total amount that Judi should have paid, we multiply 
is the bill Judi would have paid if she had money. Hence, to calculate the total amount, we multiply 
to find the total the 8 friends paid.
Solution 2
Let 
be the total bill the 8 friends split up; each person was supposed to pay 
. Since Judi forgot her money, we have to add 
to get the amount of money each of the 
friends have to pay. After multipling by 7, we have 
. This is equal to because both sides represent the total bill. Solving for :![\[\frac{7m}{8} + 17.5=m\]](http://latex.artofproblemsolving.com/5/f/e/5fea9c6b062c89444126581217039d7d1f08b5f6.png)
![\[\frac{m}{8}=17.5\]](http://latex.artofproblemsolving.com/2/7/4/27427a812050391d5409e36ce50f876e749f5b84.png)
![\[m=\boxed{\textbf{(C) } 140}.\]](http://latex.artofproblemsolving.com/4/9/8/498679b9483541417e04504b2eeb2b133a6b0594.png)
~BananaBall00 ~joyoforigami (TOGFA mathematics) edits by ~megaboy6679
Video Solution
https://youtu.be/lvaWrFwW6BM ~savannahsolver
See Also
| 2013 AMC 8 (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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
