Difference between revisions of "2006 AMC 8 Problems/Problem 16"
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<math> \textbf{(A)}\ 6400\qquad\textbf{(B)}\ 6600\qquad\textbf{(C)}\ 6800\qquad\textbf{(D)}\ 7000\qquad\textbf{(E)}\ 7200 </math> | <math> \textbf{(A)}\ 6400\qquad\textbf{(B)}\ 6600\qquad\textbf{(C)}\ 6800\qquad\textbf{(D)}\ 7000\qquad\textbf{(E)}\ 7200 </math> | ||
− | + | ==Solution== | |
+ | The amount of pages Bob, Chandra, and Alice will read is in the ratio 4:6:9. Therefore, Bob, Chandra, and Alice read 160, 240, and 360 pages respectively. They would also be reading for the same amount of time because the ratio of the pages read was based on the time it takes each of them to read a page. Therefore, the amount of seconds each person reads is simply <math>160 \cdot 45 = \boxed{\textbf{(E)}\ 7200}</math>. | ||
+ | |||
+ | ==Solution 2== | ||
+ | We can solve this using equations. Let Bob read <math>x</math> pages and Chandra read <math>y</math> pages. Then Alice would read <math>760-x-y</math> pages. Then we can set up the equations using the time it takes them to read a page: <math>45x=30y=20(760-x-y)</math>. Since we only have two variables, we only need two equations. Let's use <math>45x=30y</math> and <math>45x=20(760-x-y)</math>. We can simplify <math>45x=30y</math> down to <math>3x=2y</math> and <math>45x=20(760-x-y)</math> down to <math>9x=4(760-x-y)</math>. We can then solve the equations accordingly: | ||
+ | |||
+ | <cmath>9x=3040-4x-4y</cmath> | ||
+ | <cmath>13x=3040-4y</cmath> | ||
+ | |||
+ | From the equation <math>3x=2y</math>, we know that <math>4y=6x</math>. | ||
+ | |||
+ | <cmath>13x=3040-6x</cmath> | ||
+ | <cmath>19x=3040</cmath> | ||
+ | <cmath>x=160</cmath> | ||
+ | |||
+ | Now that we know <math>x</math>, we can multiply it by <math>45</math> (Bob's speed) to get the answer: <math>160\times45</math>=<math>\boxed{\textbf{(E)}\ 7200}</math> | ||
+ | |||
+ | ~Trex226 | ||
+ | |||
+ | ==See Also== | ||
{{AMC8 box|year=2006|num-b=15|num-a=17}} | {{AMC8 box|year=2006|num-b=15|num-a=17}} | ||
+ | {{MAA Notice}} |
Latest revision as of 19:12, 11 January 2023
Contents
Problem
Problems 14, 15 and 16 involve Mrs. Reed's English assignment.
A Novel Assignment
The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds.
Before Chandra and Bob start reading, Alice says she would like to team read with them. If they divide the book into three sections so that each reads for the same length of time, how many seconds will each have to read?
Solution
The amount of pages Bob, Chandra, and Alice will read is in the ratio 4:6:9. Therefore, Bob, Chandra, and Alice read 160, 240, and 360 pages respectively. They would also be reading for the same amount of time because the ratio of the pages read was based on the time it takes each of them to read a page. Therefore, the amount of seconds each person reads is simply .
Solution 2
We can solve this using equations. Let Bob read pages and Chandra read
pages. Then Alice would read
pages. Then we can set up the equations using the time it takes them to read a page:
. Since we only have two variables, we only need two equations. Let's use
and
. We can simplify
down to
and
down to
. We can then solve the equations accordingly:
From the equation , we know that
.
Now that we know , we can multiply it by
(Bob's speed) to get the answer:
=
~Trex226
See Also
2006 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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.