2003 AMC 10A Problems/Problem 13
Contents
Problem
The sum of three numbers is . The first is four times the sum of the other two. The second is seven times the third. What is the product of all three?
Solution
Solution 1
Let the numbers be , , and in that order. The given tells us that
Therefore, the product of all three numbers is .
Solution 2
Alternatively, we can set up the system in equation form:
Or, in matrix form
To solve this matrix equation, we can rearrange it thus:
Solving this matrix equation by using inverse matrices and matrix multiplication yields
Which means that , , and . Therefore,
Solution 3 (Simpler)
Let's denote the 3rd number as , the 2nd as , and the 1st as according to the information given in the problem. We know that all three numbers add up to , so . Solving the equation we get . Then substitute this value of x to solve for the other two numbers. Lastly, we obtain as the product of all three numbers.
Video Solution by WhyMath
~savannahsolver
See also
2003 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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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