ACT Math Practice Test Pool 1
In a particular card game, the minimum score a player can achieve in a single game is 20, and the maximum score possible in a single game is 52. If a player plays three games and scores a total of 141 points, what is the least number of points that the player could have scored in one of the games?
The least possible score in one game will occur when the player has scored the maximum possible (52 points) in each of the other two games. These two games add to 104 points, leaving 141 − 104 = 37 points as the lowest possible score for the remaining game.
What is the sum of the four times the largest negative integer and the small- est positive integer?
The largest negative integer is −1, and the smallest positive integer is 1, so the sum is 4(−1) + 1 = −4 + 1 = −3.
Three consecutive integers are such that four times the smallest is three times the largest. What is the largest of these three integers?
Strategy: work with the answers. Suppose the answer is C so that the largest of the three consecutive integers is 10. Then, the group of integers must be 8, 9, and 10. Four times the smallest is 32, and three times the largest is 30, so answer C is incorrect. If we use answer D, our group of integers is 10, 11, and 12, but 40 6= 36, so answer D is incorrect as well. Since answer C was closer, try going the other way and work with answer B. Here, the group is 6, 7, and 8, and 4×6 = 3×8, so answer B is the correct answer. Math Teacher Solution: Let x be the smallest of the three integers. Then, x+1 is the next integer, and x+2 is the biggest integer. We need 4x = 3(x+2), so that 4x = 3x + 6, resulting in x = 6.
Let f(x) = 4x − 3. If f(a) = 9 and f(b) = 5, then what is f(a + b) ?
Since f(a) = 4a − 3 and f(a) = 9, then 4a − 3 = 9 so that 4a = 12 and a = 3. Similarly, f(b) = 4b−3 = 5 so that 4b = 8 and b = 2. Finally, f(a+b) = f(3+2) = f(5) = 4 · 5 − 3 = 17.
Which of the following is NOT a positive multiple of 9 + 3 ?
Which of the following is true about the number 0.6666 ?
Since 0.6666 = 2/3, and any fraction with integers on the top and bottom is a rational number, answer A is correct. Why are the other answers incorrect? The number 2/3 can be correctly gridded as “.666” or “.667” or “2/3” but not as “0.66”. Also, 67/100 = 0.67 is not the same as 0.6666. Finally, 0.6666 isn’t a little devil every night of the week, so we can’t be sure about last night.
The price of a shirt is increased by 20%, and the price of a $40 pair of shoes is decreased by 40%. If the new price of the shirt now equals the new price of the shoes, what was the original price of the shirt?
First, determine the new price of the shoes: 40% of $40 is (40/100) · $40 = $16, so the new price of the shoes is $40 − $16 = $24. At this point, we can work with the answers to figure out which one is correct: increase each answer by 20% until we get $24. The correct answer is B since 20% of $20 is $4 and $20 + $4 = $24.
When a number x is subtracted from 36 and the difference is divided by x, the result is 2. What is the value of x ?
Strategy: work with the answers. Go through the answers, substituting each for x and using it in the problem until it works. For example, to check x = 4 (answer B), subtract 4 from 36 to get 32, and divide by 4, resulting in 8, so answer B is incorrect. You will find that only answer D gives you the final result of 2. Math Teacher Solution: Convert the words into an equation for x: 36 − x x = 2. Multiplying both sides by x gives 36 − x = 2x so that x = 12.
In a box containing only purple and green marshmallows, 6 marshmallows are purple. If the probability of choos- ing a purple marshmallow from the box is 1/3 , how many green marshmallows are in the box?
Strategy: work with the answers. Use the answers to try choices for the number of green marshmallows. If there are 9 green marshmallows (answer C), then there are 6 + 9 = 15 total marshmallows, so the probability of choosing a purple one is 6/15 = 2/5 > 1/3, so answer C is incorrect. We need more green marshmallows to get a lower probability for choosing a purple marshmallow, so we try an answer to the right. If there are 12 green marshmallows (answer D), then there are 6 + 12 = 18 total marshmallows, so the probability of choosing a purple one is 6/18 = 1/3. Answer D is correct. Math Teacher Solution: Let x be the number of green marshmallows. Then, 6 + x is the total number of marshmallows. The probability of choosing a purple marshmallow is the number of purple marshmallows (6) divided by the total number of marshmallows (6 + x). So, we need: 6/6+x=1/3 Cross-multiplying, 18 = 6 + x so that x = 12.
A number is divided by four. The result is divided by three, for a final result of two. What was the original number?
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