Algebra Jumble
Name
(1)
In the following equation, what is the sum of positive integers a and b?
1
7
=
22
a + b1
(2)
Given that 12a + 10b = 1020, what is the value of
(3)
Let c(a, b, c) =
a
b
+
b
c
a
5
+ b6 ?
+ ca . Compute c(2, 12, 9).
(4)
The sum of three numbers a, b, and c is 99. If we increase a by 6, decrease
b by 6 and multiply c by 5, the three resulting numbers are equal. What is the value of b?
(5)
The cost of holding a concert is the sum of the fixed cost, which is the
same no matter how many people attend, and the variable cost, which depends on the
number of people attending. If the total cost for a concert attended by 1000 people is
$75, 000 and the total cost of a concert attended by 1200 people is $85, 000, what is the
number of dollars in the fixed cost when holding a concert?
(6)
Each edge of a cube is decreased by 40%. What is the percent of decrease
in the volume of the cube? Express your answer to the nearest tenth.
(7)
A faucet leaks at a rate of 1 pint every 2 hours for 4 full days. How many
gallons of water leaked from the faucet?
(8)
A hose would fill a non-leaking empty clearwater pool in 5 hours. In Eric’s
Clearwater pool there is a leak at the bottom that would empty a full pool in 20 hours.
After two hours of trying to fill Eric’s pool with a hose, starting from empty, what percent
of the pool will be filled?
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Answer Sheet
Number
1
2
3
4
5
6
7
8
Answer
10
17
6
51
25,000 dollars
78.4 %
6
30
Problem ID
B2A4
20101
0403
B1A4
11D4
5CB3
21B3
3123
Copyright MATHCOUNTS Inc. All rights reserved
Solutions
(1) 10
ID: [B2A4]
Take the reciprocal of both sides and express the left-hand side as a mixed number to obtain
1
1
3 =a+ .
7
b
If a were greater than 3, then b would be negative, and if a were less than 3 then 1/b
would be greater than 1. Therefore, a = 3, b = 7, and a + b = 10 .
(2) 17
ID: [20101]
No solution is available at this time.
(3) 6
ID: [0403]
We have
2
12 9
+
+
12
9
2
1 4 9
= + +
6 3 2
1 8 9
= + +
6 6 2
3 9
9 9
= + = + = 6.
6 2
2 2
c(2, 12, 9) =
(4) 51
ID: [B1A4]
We are given the equations a + b + c = 99 and a + 6 = b − 6 = 5c. Solve b − 6 = 5c for b
to find b = 5c + 6, and solve 5c = a + 6 for a to find a = 5c − 6. Substituting both of
these equations into a + b + c = 99, we have (5c − 6) + (5c + 6) + c = 99. Simplifying the
left hand side, we get 11c = 99 which implies c = 9. Substituting into b = 5c + 6, we have
b = 5(9) + 6 = 51 .
(5) 25,000 dollars
ID: [11D4]
No solution is available at this time.
(6) 78.4 %
ID: [5CB3]
No solution is available at this time.
(7) 6
ID: [21B3]
No solution is available at this time.
(8) 30
ID: [3123]
No solution is available at this time.
Copyright MATHCOUNTS Inc. All rights reserved