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SOURCE:COMPETITION Number of Problems: 10. FOR PRINT ::: (Book)
Let . What is ?
Let . Since one of the sums involves a, b, c, and d, it makes sense to consider 4x. We have . Rearranging, we have , so . Thus, our answer is .
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?
Let the numbers be , , and in that order. The given tells us that
Therefore, the product of all three numbers is .
Alternatively, we can set up the system in matrix 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,
A school store sells 7 pencils and 8 notebooks for . It also sells 5 pencils and 3 notebooks for . How much do 16 pencils and 10 notebooks cost?
We let cost of pencils in cents, number of notebooks in cents. Then
Subtracting these equations yields . Backwards solving gives . Thus the answer is .
Suppose that of bananas are worth as much as oranges. How many oranges are worth as much as of bananas?
If , then .
The product of two positive numbers is 9. The reciprocal of one of these numbers is 4 times the reciprocal of the other number. What is the sum of the two numbers?
Let the two numbers equal and . From the information given in the problem, two equations can be written:
Therefore,
Replacing with in the equation,
So and would then be
The sum would be =
The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?
Let the three numbers be equal to , , and . We can now write three equations:
Adding these equations together, we get that
and
Substituting the original equations into this one, we find
Therefore, our numbers are 12, 7, and 5. The middle number is