60. Orangeade Pricing
On a certain day, orangeade was made by mixing a certain amount of orange juice with an equal amount of water. On the next day, orangeade was made by mixing the same amount of orange juice with twice the amount of water. On both days, all the orangeade that was made was sold. If the revenue from selling the orangeade was the same for both days and if the orangeade was sold at $0.60 per glass on the first day, what was the price per glass on the second day?
Solution:
Let the amount of orange juice = a. Then:- Day 1: Water = a. Total orangeade = 2a. Price per glass = $0.60. Revenue = 0.60 × (2a) = 1.2a.
- Day 2: Water = 2a. Total orangeade = 3a. Let price per glass = p. Revenue = p × (3a) = 3pa.
Revenue is the same for both days:
1.2a = 3pa
1.2 = 3p
p = 0.40
Correct Answer: D
61. Slope of a Line
In the xy-plane, what is the slope of the line with equation 3x + 7y = 9?
Solution:
Put the equation in slope-intercept form y = mx + b:7y = -3x + 9
y = (-3/7)x + 9/7
The slope is -3/7.
Correct Answer: B
63. Kilometer to Mile Conversion
If 1 kilometer is approximately 0.6 mile, which of the following best approximates the number of kilometers in 2 miles?
Solution:
Set up a proportion: 1 km/0.6 mi = k km/2 mik = 2/0.6 = 2/(3/5) = 2 × 5/3 = 10/3
Correct Answer: A
64. Fruit Stand Purchase
A certain fruit stand sold apples for $0.70 each and bananas for $0.50 each. If a customer purchased both apples and bananas from the stand for a total of $6.30, what total number of apples and bananas did the customer purchase?
Solution:
Let a = number of apples, b = number of bananas.0.70a + 0.50b = 6.30
Multiply by 10: 7a + 5b = 63
5b = 63 - 7a
b = (63 - 7a)/5
For b to be an integer, 63 - 7a must be divisible by 5. Test integer values for a:
- a=4: 63 - 28 = 35, b=7. Total = 4+7=11.
- a=9: 63 - 63 = 0, b=0. But the problem says the customer purchased both, so this is invalid.
Thus, total is 11.
Correct Answer: B
65. Distance Conversion
The average distance between the Sun and a certain planet is approximately 2.3 × 10¹⁴ inches. Which of the following is closest to the average distance between the Sun and the planet, in kilometers? (1 kilometer is approximately 3.9 × 10⁴ inches.)
Solution:
Convert inches to kilometers: Distance (km) = (2.3 × 10¹⁴)/(3.9 × 10⁴)≈ (2.3/3.9) × 10¹⁰ ≈ 0.59 × 10¹⁰ = 5.9 × 10⁹
Correct Answer: B
66. School Grade Ratios
At a certain school, the ratio of the number of second graders to the number of fourth graders is 8 to 5, and the ratio of the number of first graders to the number of second graders is 3 to 4. If the ratio of the number of third graders to the number of fourth graders is 3 to 2, what is the ratio of the number of first graders to the number of third graders?
Solution:
Assign variables based on the ratios:- 2nd : 4th = 8:5 → Let 2nd = 8a, 4th = 5a
- 1st : 2nd = 3:4 → 1st/8a = 3/4 → 1st = 6a
- 3rd : 4th = 3:2 → 3rd/5a = 3/2 → 3rd = 7.5a = 15a/2
Find 1st : 3rd = 6a : (15a/2) = 6 : 15/2 = 12 : 15 = 4 : 5
Correct Answer: E
67. Multiples of 5
If m is the average (arithmetic mean) of the first 10 positive multiples of 5 and if M is the median of the first 10 positive multiples of 5, what is the value of M - m?
Solution:
The first 10 positive multiples of 5 are: 5, 10, 15, ..., 50. This is an arithmetic sequence.- Average m = (first + last)/2 = (5 + 50)/2 = 55/2 = 27.5
- Median M = average of 5th and 6th terms = (25 + 30)/2 = 55/2 = 27.5
Thus, M - m = 0.
Correct Answer: B
68. Probability of Sum
Two integers will be randomly selected from the sets above, one integer from set A and one integer from set B. What is the probability that the sum of the two integers will equal 9?
A = {2, 3, 4, 5}
B = {4, 5, 6, 7, 8}
A = {2, 3, 4, 5}
B = {4, 5, 6, 7, 8}
Solution:
Total number of outcomes: 4 × 5 = 20.Pairs that sum to 9: (2,7), (3,6), (4,5), (5,4). There are 4 favorable outcomes.
Probability = 4/20 = 0.20.
Correct Answer: B
69. Circle Area
In the coordinate plane, a circle has center (2,-3) and passes through the point (5,0). What is the area of the circle?
Solution:
The radius is the distance between the center (2, -3) and the point (5, 0).r = √[(5-2)² + (0-(-3))²] = √[3² + 3²] = √18 = 3√2
Area = πr² = π(3√2)² = π × 18 = 18π
Correct Answer: E
70. Highway Traffic Estimation
At a certain instant in time, the number of cars, N, traveling on a portion of a certain highway can be estimated by the formula
N = (20Ld)/(600 + s²)
where L is the number of lanes in the same direction, d is the length of the portion of the highway, in feet, and s is the average speed of the cars, in miles per hour. Based on the formula, what is the estimated number of cars traveling on a ½-mile portion of the highway if the highway has 2 lanes in the same direction and the average speed of the cars is 40 miles per hour? (5,280 feet = 1 mile)
N = (20Ld)/(600 + s²)
where L is the number of lanes in the same direction, d is the length of the portion of the highway, in feet, and s is the average speed of the cars, in miles per hour. Based on the formula, what is the estimated number of cars traveling on a ½-mile portion of the highway if the highway has 2 lanes in the same direction and the average speed of the cars is 40 miles per hour? (5,280 feet = 1 mile)
Solution:
L = 2, s = 40, d = 0.5 mile = 0.5 × 5280 = 2640 feetN = (20 × 2 × 2640)/(600 + 40²) = 105600/(600 + 1600) = 105600/2200 = 48
Correct Answer: D
71. Stock Price Changes
Yesterday's closing prices of 2,420 different stocks listed on a certain stock exchange were all different from today's closing prices. The number of stocks that closed at a higher price today than yesterday was 20 percent greater than the number that closed at a lower price. How many of the stocks closed at a higher price today than yesterday?
Solution:
Let L = number that closed lower. Then H = number that closed higher = 1.2L.Since all prices changed, H + L = 2420.
1.2L + L = 2420
2.2L = 2420
L = 1100
Thus, H = 1.2 × 1100 = 1320.
Correct Answer: D
72. Solving for x
If y((3x - 5)/2) = y and y ≠ 0, then x =
Solution:
Since y ≠ 0, we can divide both sides by y:(3x - 5)/2 = 1
3x - 5 = 2
3x = 7
x = 7/3
Correct Answer: C
73. Inequality Range
If x + 5 > 2 and x - 3 < 7, the value of x must be between which of the following pairs of numbers?
Solution:
Solve the inequalities:x + 5 > 2 ⇒ x > -3
x - 3 < 7 ⇒ x < 10
So, -3 < x < 10.
Correct Answer: A
74. LCM Problem
A gym class can be divided into 8 teams with an equal number of players on each team or into 12 teams with an equal number of players on each team. What is the lowest possible number of students in the class?
Solution:
The number of students must be divisible by both 8 and 12. The least common multiple (LCM) of 8 and 12 is 24.
Correct Answer: B
76. Committee Voting
At least 2/3 of the 40 members of a committee must vote in favor of a resolution for it to pass. What is the greatest number of members who could vote against the resolution and still have it pass?
Solution:
The minimum number to pass is (2/3) × 40 = 26.66..., so at least 27 votes are needed.Therefore, the maximum number against is 40 - 27 = 13.
Correct Answer: E
77. Factorial Divisibility
If n = 20! + 17, then n is divisible by which of the following?
I. 15
II. 17
III. 19
I. 15
II. 17
III. 19
Solution:
- 20! is divisible by all integers from 1 to 20.- 20! is divisible by 15, so 20! + 17 is 17 more than a multiple of 15, so it is not divisible by 15.
- 20! is divisible by 17, so 20! + 17 is 17 more than a multiple of 17, so it is divisible by 17.
- 20! is divisible by 19, so 20! + 17 is 17 more than a multiple of 19, so it is not divisible by 19.
Thus, only II is true.
Correct Answer: C
79. Running Distance
After driving to a riverfront parking lot, Bob plans to run south along the river, turn around, and return to the parking lot, running north along the same path. After running 3.25 miles south, he decides to run for only 50 minutes more. If Bob runs at a constant rate of 8 minutes per mile, how many miles farther south can he run and still be able to return to the parking lot in 50 minutes?
Solution:
Bob's speed = 1/8 mile per minute.Let x be the additional miles he runs south. Then he must run x + 3.25 miles north to return.
Total running time for the rest of the trip = time south + time north = 8x + 8(x + 3.25) = 16x + 26 minutes.
This must equal 50 minutes: 16x + 26 = 50
16x = 24
x = 1.5
Correct Answer: A
80. Sum of Reciprocals
M is the sum of the reciprocals of the consecutive integers from 201 to 300, inclusive. Which of the following is true?
Solution:
There are 100 terms in the sum.The smallest term is 1/300, the largest is 1/201.
So, 100 × (1/300) < M < 100 × (1/201)
1/3 < M < 100/201
Since 100/201 ≈ 0.4975 < 1/2, we have 1/3 < M < 1/2.
Correct Answer: A
81. Machine Rates
Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours. Working alone at its constant rate, Machine A produces 800 nails in y hours. In terms of x and y, how many hours does it take Machine B, working alone at its constant rate, to produce 800 nails?
Solution:
Let r_A and r_B be the rates of A and B in nails/hour.Together: r_A + r_B = 800/x
A alone: r_A = 800/y
So, r_B = 800/x - 800/y = 800(1/x - 1/y) = 800((y-x)/xy)
Time for B alone = 800 / r_B = 800 / [800((y-x)/xy)] = xy/(y-x)
Correct Answer: E
82. Budget Allocation
In the Johnsons' monthly budget, the dollar amounts allocated to household expenses, food, and miscellaneous items are in the ratio 5:2:1, respectively. If the total amount allocated to these three categories is $1,800, what is the amount allocated to food?
Solution:
Total ratio parts = 5 + 2 + 1 = 8.Value of one part = 1800 / 8 = 225.
Food is 2 parts: 2 × 225 = 450.
Correct Answer: D
83. Board Members
There are 4 more women than men on Centerville's board of education. If there are 10 members on the board, how many are women?
Solution:
Let m = number of men. Then w = m + 4.m + (m+4) = 10
2m + 4 = 10
2m = 6
m = 3
w = 3 + 4 = 7
Correct Answer: D
84. Compound Interest
Leona bought a 1-year, $10,000 certificate of deposit that paid interest at an annual rate of 8 percent compounded semiannually. What was the total amount of interest paid on this certificate at maturity?
Solution:
Semiannual interest rate = 8%/2 = 4%.After first 6 months: Value = 10000 × 1.04 = 10400.
After second 6 months: Value = 10400 × 1.04 = 10816.
Interest = 10816 - 10000 = 816.
Correct Answer: C
85. Decimal Calculation
[(0.0036)(2.8)]/[(0.04)(0.1)(0.003)] =
Solution:
[(0.0036)(2.8)]/[(0.04)(0.1)(0.003)] = [(36 × 10⁻⁴)(28 × 10⁻¹)]/[(4 × 10⁻²)(1 × 10⁻¹)(3 × 10⁻³)]= [(36 × 28) × 10⁻⁵]/[(4 × 1 × 3) × 10⁻⁶] = (1008 × 10⁻⁵)/(12 × 10⁻⁶) = 84 × 10¹ = 840
Correct Answer: A
86. Machine Production
Machine A produces bolts at a uniform rate of 120 every 40 seconds, and Machine B produces bolts at a uniform rate of 100 every 20 seconds. If the two machines run simultaneously, how many seconds will it take for them to produce a total of 200 bolts?
Solution:
Rate of A = 120/40 = 3 bolts/second.Rate of B = 100/20 = 5 bolts/second.
Combined rate = 3 + 5 = 8 bolts/second.
Time for 200 bolts = 200 / 8 = 25 seconds.
Correct Answer: B
87. Divisibility by 3
If n is an integer greater than 6, which of the following must be divisible by 3?
Solution:
For a product to be divisible by 3, at least one factor must be divisible by 3.Consider remainders modulo 3. The numbers n, n+1, n+2 form a complete residue system modulo 3. One of them is divisible by 3.
Check each option to see if the set of three factors always contains one of n, n+1, n+2.
(A) n, n+1, n-4 ≡ n, n+1, n+2 (mod 3). This set always contains a multiple of 3.
The others can be checked and will fail for some n.
Correct Answer: A
88. Gross Profit
The total cost for Company X to produce a batch of tools is $10,000 plus $3 per tool. Each tool sells for $8. The gross profit earned from producing and selling these tools is the total income from sales minus the total production cost. If a batch of 20,000 tools is produced and sold, then Company X's gross profit per tool is
Solution:
Total Cost = 10000 + 3(20000) = 10000 + 60000 = 70000.Total Revenue = 8 × 20000 = 160000.
Gross Profit = 160000 - 70000 = 90000.
Gross Profit per tool = 90000 / 20000 = 4.50.
Correct Answer: C
89. Battery Pricing
A dealer originally bought 100 identical batteries at a total cost of q dollars. If each battery was sold at 50 percent above the original cost per battery, then, in terms of q, for how many dollars was each battery sold?
Solution:
Cost per battery = q/100.Selling price per battery = 1.5 × (q/100) = (3/2)(q/100) = 3q/200.
Correct Answer: A
90. Consecutive Integers
In an increasing sequence of 10 consecutive integers, the sum of the first 5 integers is 560. What is the sum of the last 5 integers in the sequence?
Solution:
Let the first integer be x.The first 5 integers: x, x+1, x+2, x+3, x+4. Their sum is 5x + 10 = 560.
5x = 550
x = 110
The last 5 integers are x+5, x+6, x+7, x+8, x+9.
Their sum is 5x + (5+6+7+8+9) = 5x + 35.
Substitute x=110: 5(110) + 35 = 550 + 35 = 585.
Correct Answer: A