Download Algebra Review March 31, 2014 Cascade Ridge PTSA Math Club 1

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Algebra Review
March 31, 2014
1. Gale and Katniss each shot 6 arrows at the target shown. Gale shot 4 arrows into A and 2 into B;
his score was 18 points. Katniss shot 3 into A and 3 into B; her score was 21 points. How many
points were given for an arrow in B?
A
B
2. The sum of the heights of Paul and Rachel is 94 inches. Rachel is 8 inches taller than Paul. How
many inches tall is Paul?
3. The sum of the ages of Alice, Betty, and Clara is 29 years. Betty is 4 years older than Alice and
Clara is 6 years older than Betty. What is Alice’s age?
4. The quotient of two numbers is 4 and their difference is 39. What is the smaller number of the
two?
5. Patricia has $12 more than Rhoda and $15 more than Sarah. Together all three have $87. How
much does Patricia have?
6. N is the number of buttons in a sewing box.
N is more than 40 but less than 80.
When N is divided by 5, the remainder is 2.
When N is divided by 7, the remainder is 4.
Find the value of N.
7. Barbara has 20 coins consisting of nickels and dimes. If the nickels were dimes and the dimes
were nickels, she would have 30 cents more than she has now. How many dimes did she have
to begin with?
8. A restaurant has a total of 30 tables which are of two types. The first type seats two people at
each table; the second type seats five people at each table. A total of 81 people are seated
when all seats are occupied. How many tables for two are there?
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Algebra Review
March 31, 2014
BONUS PROBLEMS
9. Peter agreed to work after school for 8 weeks at a fixed weekly rate. But instead of being given
only money, he was to be given $85 and a bicycle. However, Peter worked only 5 weeks at the
fixed weekly rate and was given $25 and the bicycle. How much was the bicycle worth?
10. The ratio of Ken’s age to his mother’s age is 2:7. In 3 years, the sum of their ages will be 60.
How old is each now?
11. If a class of children is divided up into groups of 5 children, 2 children will be left over. If the
class is divided into groups of 6 children, 3 children will be left over. What is the smallest
number of children the class could have?
12. Two cars drive toward each other from opposite ends of a highway 767 miles long. The speed of
the first car is 56 miles per hour. The speed of the second car is 62 miles per hour. The cars will
meet after how many hours of driving?
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Algebra Review
March 31, 2014
SOLUTIONS:
1. 5
We know that 4A + 2B = 18 and 3A + 3B = 21. If we divide both sides of the first equation by 2
and both sides of the second equation by 3, we get 2A + B = 9 and A + B = 7. Subtracting the
second equation from the first, we get A = 2, which means that B = 5.
2. 43 inches
The average height of the two people is 94/2 = 47 inches. Since the people differ in height by 8
inches, each differs from the average height by 4 inches. Then Rachel is 47 + 4 = 51 inches tall
and Paul is 47 – 4 = 43 inches tall. (Note: This method can be used whenever the sum and
difference of two numbers is known.)
3. 5 years old
A + B + C = 29. A + 4 = B and B + 6 = C. Substituting A + 4 for B, we have:
A + A + 4 + A + 4 + 6 = 29
3A + 14 = 29
3A = 15
A=5
4. 13
Let x = the smaller number. Then the larger number is 4x. 4x – x = 39, so 3x = 39 and x = 13.
5. $38
If we add $12 to what Rhoda has, she will have as much as Patricia. If we add $15 to what Sarah
has, she will also have as much as Patricia. Since all three together have $87, then $87 + $12 +
$15 or $114 represents three times what Patricia has. Thus Patricia has $114/3 or $38.
6. N = 67
The number N must end in 2 or 7. Begin with 6 x 7 + 4 = 46 and count by 7s to the first number
ending in 2 or 7. This yields 46, 53, 60, 67. The required number is 67.
7. 7 dimes
N + D = 20. 5N + 10D + 30 = 10N + 5D. Combining like terms: 5N – 5D = 30. Dividing both sides
by 5: N – D = 6. Adding the two equations together: 2N = 26, so N = 13 and D = 7.
8. 23 tables
Suppose just 2 people were seated at each of the 30 tables. Then a total of 60 people would be
seated. Since the seating capacity is 81, there would be 21 vacant seats. Each of the tables for 5
people has 3 vacant seats. There must be 7 tables that seat 5 people. Thus there must be 23
tables that seat 2 people.
9. $75
When Peter worked 3 fewer weeks than the agreed period of 8 weeks, he received $60 less than
the original amount. Therefore each week’s work yielded $20. At this rate, he was entitled to
$160 for 8 weeks. Since he was to receive $85 and a bicycle, the bicycle was worth 160 – 85 =
$75.
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Algebra Review
March 31, 2014
10. Ken is 12 and his mother is 42.
2M = 7K
M + 3 + K + 3 = 60.
M + K = 54; K = 54 – M
2M = 7(54 – M) = 378 – 7M
9M = 378
M = 42
84 = 7K, K = 12
11. 27
Each multiple of 5, increased by 2, ends either in 2 or 7. Each multiple of 6, increased by 3, must
be an odd number. Then the only possibilities for the class size are: 7, 17, 27, 37,… . Divide each
by 6. The first number that has a remainder of 3 is the required number. That number is 27.
12. 6.5 hours
Remembering that distance = rate · time, the total distance both cars will have traveled together
is 767 miles. So 56t + 62t = 767; 118t = 767, t = 6.5 hours.
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