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SAT Test Prep : Math Multiple-Choice I

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Choose the best answer from the five choices given.

1. Tom has 7 times as many crayons as Dick and 3 times as many as Harry. If Dick has less than 21 crayons, what is the maximum number of crayons that Tom can have?
• 63
• 105
• 120
• 140
• 147
Answer: 105
Tom has 7 times as many crayons as Dick and 3 times as many as Harry.
So, the number of crayons Tom has must be a multiple of 21 (i.e., 7 × 3).
Therefore, 140 and 120 are not possible options.
Also, if Tom has 147 crayons, then Dick will have 21 crayons; but, Dick has less than 21 crayons.
So, the maximum number of crayons that Tom can have is 105.


2. What is the smallest prime factor of 1001?
• 3
• 5
• 7
• 11
• 13
Answer: 7
Now, 1001 is divisible by neither 3 nor 5.
So, let's try to divide by 7. We find 1001 / 7 = 143 = 13 × 11.
Since 1001 = 13 × 11 × 7, the smallest prime factor is 7.


3. The arithmetic mean (average) of 3, 5, 7 and y is 100. What is the value of y?
• 85
• 95
• 105
• 115
• 385
Answer: 385
The arithmetic mean of n numbers is their sum divided by n.
So, (3 + 5 + 7 + y) / 4 = 100 or 15 + y = 100 × 4 = 400.
Thus, y = 400 − 15 = 385.
Given that 3, 5 and 7 are small numbers and the average 100 is relatively large, it is possible to guess the answer as 385.



4. A quiz consists of true and false questions. The ratio of the number of true questions to the number of false questions is 4:3. About what percent of the questions are false?
• 43%
• 57%
• 67%
• 83%
• 86%
Answer: 43%
Percent of false questions = (3/7) × 100 = 42.86.
So, about 43% of the questions are false.
Alternatively, one could reason out without any calculations as follows.
For every 4 true questions, there are 3 false questions. So, there are fewer false questions than true ones, i.e., less than 50% of the questions are
false. The only option is 43%.


5. A small square region inside a larger square is shaded. The diagonal of the larger square is three times the diagonal of the small square in length. What fraction of the area of the larger square is not shaded ?
• 1/9
• 1/3
• 2/3
• 8/9
• cannot be determined
Answer: 8/9
Let b be the side of the square and d the length of its diagonal.
Then, by Pythagoras theorem, d2 = b2 + b2 = 2 b2.
Since the diagonal of the larger square is 3 times that of the small square,
the side of the larger square is 3 times the side of the small square.
If b is the side of the small shaded square, then its area = b2.
Now, 3b is the side of the larger square and its area = 9b2.
Thus, unshaded area = 8b2 and
fraction of the area of the larger square unshaded = 8/9.


6. An agent receives a commission of 40 cents for every $50 of business she procures. What percent is the agent's commission?
• 0.8%
• 1.0%
• 1.25%
• 1.5%
• 2.5%
Answer: 0.8%
Percent Commission = (0.40/50) × 100 = 40/50 = 0.8.
Alternatively, 1% of $50 is 50 cents. Since the commission is 40 cents, it is less than 1%. So, the only option is 0.8%.


7. What is 2% of 7%?
• 0.014%
• 0.09%
• 0.14%
• 1.4%
• 14%
Answer: 0.14%
2% of 7 = (2/100) × 7 = 14/100 = 0.14. So, 2% of 7% = 0.14%.


8. What is z if 2.5% of z equals 75% of 50?
• 10
• 25
• 50
• 75
• 1500
Answer: 1500
Now, (2.5/100) z = (75/100) 50. So, z = (75/2.5) 50 = (750/25) 50 = 1500.
Alternatively, 75 is 30 times 2.5. So, z must be 30 times 50.
Of the given options, only 1500 is large enough.
To save time, you can sometimes guess the answer without doing detailed computations.


9. What is the smallest of four consecutive odd integers whose sum is 968?
• 235
• 239
• 243
• 245
• 249
Answer: 239
Let the smallest of the four consecutive odd integers be m. Then,
m + (m + 2) + (m + 4) + (m + 6) = 968 or 4m + 12 = 968.
Thus, m = 956 / 4 = 239.
The four consecutive odd integers are 239, 241, 243 and 245.


10. A car's price decreased by 20% from 1998 to 2000. If the price of the car was P dollars in 2000, what was it in 1998?
• 0.75P
• 0.8P
• 1.2P
• 1.4P
• 1.25P
Answer: 1.25P
Let the car's price in 1998 be x dollars.
Then, a 20% price decrease implies the price in 2000 was (1 − 0.2)x.
So, P=0.8x = 4x/5 or x = 5P/4 = 1.25P.


11. Which of the following is not equivalent to 4/5?
• 40/50
• 80/100
• 400/500
• 0.8
• 40%
Answer: 40%
40% = 40/100 = 4/10 = 2/5, whereas 400/500 = 40/50 = 80/100 = 0.8 = 4/5.


12. An office requires P sheets of paper each month for each employee. If there are E employees in the office, for how many months willT sheets of paper last?
PE/T
TPE
TE/P
T/PE
P/TE
Answer: T/PE
The office requires PE sheets per month.
Therefore, the T sheets of paper will last T/PE months.
If the problem appears confusing, choose an easy set of numbers,
e.g., P = 30; E = 10 and T = 600.
Then, the office requires 300 sheets (i.e., 30 × 10) per month.
Therefore, the 600 sheets of paper will last 2 months (i.e., 600/300).


13. If x + yz = 5 and xy + z = 10, which of the following statements MUST be true?
I. y > z
II. x > 5
III. y > 5
• I only
• II only
• III only
• I and II only
• II and III only
Answer: II only
Adding the equations gives 2x = 15 or x = 7.5. So, x > 5 (II is true).
Substituting x = 7.5 gives zy = 2.5. So, z > y (I is false).
There are no further conditions on y. So, III is not necessarily true.


14. From 1999 to 2000, the computer sales increased by 40% and the printer sales decreased by 40%. The ratio of printer sales to computer sales in 2000 was how many times the ratio of printer sales to computer sales in 1999?
• 3/5
• 1
• 3/7
• 5/3
• 7/3
Answer: 3/7
Let c be the computer sales in 1999 and p the printer sales in 1999.
Then, the computer sales in 2000 are 1.4c and the printer sales in 2000 are 0.6p.
Ratio of printer sales to computer sales in 2000 = 0.6p/1.4c.
Ratio of printer sales to computer sales in 1999 = p/c.
Dividing the above two ratios, we get 0.6/1.4 = 6/14 = 3/7.


15. A toy factory manufactures d dolls every hour. Each doll costs c cents. How many dollars will the factory spend in costs for manufacturing dolls in 7 hours and 30 minutes?
cd/100
• 3cd/40
• 7cd/100
cd/750
• 15cd/100
Answer: 3cd/40
Manufacturing Cost = (d dolls/hour) (c cents/doll)
= cd cents/hour = cd/100 $/hr.
Now, 7 hours and 30 minutes = 7 1/2 hours = 15/2 hours.
Total Manufacturing Cost = (cd/100 $/hr) (15/2 hr) = 15cd/200 $ = 3cd/40 $.


16. If 2z2 − 5z + 30 < 2z2 + 5z − 15, which of the following MUST be true?
z = 4.5
z > 4.5
z < 4.5
z > 6
z = 6
Answer: z > 4.5
The given inequality can be simplified as follows.
2z2 + 5z − 15 − 2z2 + 5z − 30 > 0
10z − 45 > 0 or 10z > 45 or z > 4.5


17. An total of $45,000 in prize money is to be divided unequally among the top three contestants. What is the value of the largest prize?
• $9,000
• 0,000
• 2,000
• 5,000
• $27,000
Answer: $27,000
If $45,000 is divided equally among the three, then each will get 5,000. Since the prize money is to be divided unequally, the largest prize will be more than 5,000. The only option is $27,000.


18. The arithmetic mean (average) of five numbers is −5. If the sum of two of them is 50, what is the average of the other three?
• 25
• 10
• 0
• −10
• −25
Answer: −25
The arithmetic mean of n numbers is their sum divided by n.
So, Sum of five numbers / 5 = −5 or Sum of five numbers = −25.
Now, Sum of three numbers + 50 = −25 or Sum of three numbers = −75.
Thus, Average = Sum of three numbers / 3 = −75/3 = −25.
Here is an alternative strategy:
Given that the average of all the five numbers is negative (−5), their sum is negative. Given that the sum of two of them is 50 (positive), the sum and
average of the other three must be negative. So, the only options are −10 and −25.
Check for −10: Average of five numbers = (50 + 3 (−10))/5 = 4. So, incorrect.
Check for −25: Average of five numbers = (50 + 3 (−25))/5 = −5. So, correct.


19. In the figure, each of the four white "petals" is formed by the intersection of two semicircles. Each of the four semicircles is drawn with its center at the midpoint of each side of the square. If the length of the side of the square is 2, what is the total area of the shaded region excluding the petals?
• 8 − 2π
• 4 − π
• 1 − π/4
• 16 − 4π
• 32 − 8π
Answer: 8 − 2π
Consider one-fourth of the square with half a semicircle as shown in the figure.
Area of one-fourth of the square = 1 × 1 = 1.
Area of half a semicircle = Area of circle / 4 = π(radius)2/4 = π(1)2/4 = π/4.
Area of shaded portion in figure alongside = 1 − π/4.
There are eight such shaded portions in the complete square.
So, total area of shaded region in the complete square = 8 − 2π.


20. Jack is now 7 times as old as Jill. If 10 years from now, he will be 3 times as old as Jill, how old is Jack now?
• 14
• 21
• 28
• 35
• 42
Answer: 35
Let Jill be y years old. So, Jack is 7y years old now.
Ten years from now, Jill will be (y + 10) and Jack will be (7y + 10).
So, 7y + 10 = 3 (y + 10) = 3y + 30.
Thus, 4y = 20 or y = 5. Jill is 5 years old and Jack is 35 years old now.


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