SAT Math Questions: Quiz! Trivia

1. If (a-b)^2 = (a+b)^2, what is the value of ab?

-4

-2

0

2

4

(a+b)^2 = (a-b)^2
a^2 + 2 ab + b^2 = a^2 - 2 ab + b^2
2 ab = -2 ab
4 ab = 0
ab = 0

Explanation

(a+b)^2 = (a-b)^2
a^2 + 2 ab + b^2 = a^2 - 2 ab + b^2
2 ab = -2 ab
4 ab = 0
ab = 0

2. How many numbers less than 1000 are divisble by 3?

300

310

311

333

500

Every multiple of three is divisible by 3, so divide 1000 by 3 and take the whole part = 333.3333 = 333

Explanation

Every multiple of three is divisible by 3, so divide 1000 by 3 and take the whole part = 333.3333 = 333

3. If a^b = c^d, which of the following is not necessarily true?

A^b - c^d = 0

A^b + c^d = 2 * (a^b)

(a^b)/(c^d) = 1

A = c

A^b * c^d = a^(2*b)

A, B, C, and E can be proven by substituting a^b for c^d. However, D is not always true. For example, 2^3 = 8^1, but 2 is not equal to 8.

Explanation

A, B, C, and E can be proven by substituting a^b for c^d. However, D is not always true. For example, 2^3 = 8^1, but 2 is not equal to 8.

4. If a + b = y, what is a^2 + 2ab + b^2?

Y

2y

Y^2

2y^2

4y^2

a^2+2ab+b^2 = (a+b)^2 = y^2

Explanation

a^2+2ab+b^2 = (a+b)^2 = y^2

5. If 2 ^ (4x + 3) = 4 ^ (x - 1), what is x?

-3

-5/2

-1

2

4/3

If a ^ b = a ^ c, then b = c. However, 2 does not equal 4, so we should make the two equal. 4 = 2 ^ 2, so 4 ^ (x - 1) = (2 ^ 2) ^ (x - 1) = 2 ^ 2(x-1) by exponent rules. So since 2 = 2, 4x + 3 = 2(x-1) -> 4x + 3 = 2x - 2 -> 2x = - 5 -> x = -5/2

Explanation

If a ^ b = a ^ c, then b = c. However, 2 does not equal 4, so we should make the two equal. 4 = 2 ^ 2, so 4 ^ (x - 1) = (2 ^ 2) ^ (x - 1) = 2 ^ 2(x-1) by exponent rules. So since 2 = 2, 4x + 3 = 2(x-1) -> 4x + 3 = 2x - 2 -> 2x = - 5 -> x = -5/2

6. A square, X, has sides of length n. Another square, Y, has sides of length 1.5n. How many X can fit into a single Y?

1

1.5

2

2.25

4

The number of X's that fit into Y is equal to the ratio of area of Y to that of X. Area of Y = (1.5n)^2 = 2.25. The area of X = n^2. 2.25n^2/n^2 = 2.25

Explanation

The number of X's that fit into Y is equal to the ratio of area of Y to that of X. Area of Y = (1.5n)^2 = 2.25. The area of X = n^2. 2.25n^2/n^2 = 2.25

7. A cubic box, X, has sides of length n. Another cubic box, Y, has sides of length 2n. How many boxes X could fit into a single box Y?

2

4

8

16

32

The volume of Y is equal to (2n)^3 = 8n^3. The volume of X is equal to n^3. So, to find the number of X's that could fit into Y, you should divide the volume of Y by that of X, or 8n^3/n^3, which is equal to 8.

Explanation

The volume of Y is equal to (2n)^3 = 8n^3. The volume of X is equal to n^3. So, to find the number of X's that could fit into Y, you should divide the volume of Y by that of X, or 8n^3/n^3, which is equal to 8.

8. A triangle has sides of length 7, 11, and X. Which of the following cannot be X?

2

4

8

12

18

The sum of lengths of two sides of a triangle cannot be less than the length of the third side. So, 2 + 7 = 9, but 9 < 11

Explanation

The sum of lengths of two sides of a triangle cannot be less than the length of the third side. So, 2 + 7 = 9, but 9 < 11

9. Which of the following lines does not intersect y = 5 x + 2?

-5x + 2y = 4

-2x + 5y = -3

10x - y = 1

3x + y = 17

5x - y = -29

You are looking for a line whose slope is equal to the slope of the given line, or 5. Slope in standard form = -A/B, so E is the answer (-5/-1 = 5 = slope)

Explanation

You are looking for a line whose slope is equal to the slope of the given line, or 5. Slope in standard form = -A/B, so E is the answer (-5/-1 = 5 = slope)

10. If A $ B = A * B - ( A + B), what is 3 $ (2 $ 1)?

-5

-3

0

1

4

First, use the order of operations to find the parenthetical value of (2 $ 1), which is 2*1-3 or -1

Next, plug in -1 for (2 $ 1) to find 3 $ -1 = 3 *-1 - (3 - 1) = -3 - 2 = -5

Explanation

First, use the order of operations to find the parenthetical value of (2 $ 1), which is 2*1-3 or -1

Next, plug in -1 for (2 $ 1) to find 3 $ -1 = 3 *-1 - (3 - 1) = -3 - 2 = -5

11. In physics, force = mass * acceleration. Suppose you have an original force F and new force G in which the mass is increased by a factor of two and the acceleration is increased by a factor of four. What is the ratio of G:F?

1:8

1:4

1:1

4:1

8:1

F = ma, G = (2m)(4a) = 8ma; 8ma:ma simplifies to 8:1

Explanation

F = ma, G = (2m)(4a) = 8ma; 8ma:ma simplifies to 8:1

12. A regular polygon has 9 sides. What is the degree measure of the angle, within the polygon, between any two sides?

60

90

120

140

165

First, you must figure the sum of the measures of the internal angles, which is equal to (n-2)180 or (9-2)180 = 7 * 180. Then, you should divide this measure by the number of sides to find the length of each angle. (7 * 180)/9 = 7 * (180/9) = 7 * 20 = 140 degrees

Explanation

First, you must figure the sum of the measures of the internal angles, which is equal to (n-2)180 or (9-2)180 = 7 * 180. Then, you should divide this measure by the number of sides to find the length of each angle. (7 * 180)/9 = 7 * (180/9) = 7 * 20 = 140 degrees

13. Six children sit at a circular table. In how many orders can they sit at the table?

6

18

64

118

120

At a circular (ring) table, the order ABCDEF = BCDEFA = CDEFAB, and so on. So, you need to find the total number of orders and divide this number by 6 to recognize the fact that the table is circular. The total number of orders is 6!, and 6!/6 = 5! = 5*4*3*2*1 = 120.

Explanation

At a circular (ring) table, the order ABCDEF = BCDEFA = CDEFAB, and so on. So, you need to find the total number of orders and divide this number by 6 to recognize the fact that the table is circular. The total number of orders is 6!, and 6!/6 = 5! = 5*4*3*2*1 = 120.

14. Which of the following values of x is not in the domain of the function y = x / (x^2-2x+1)

-3

-2

-1

0

1

Any time the denominator is equal to 0, that value of x is not included in the functional value. So, x / (x^2-2x+1) = x / (x-1)^2. (x-1)^2 = 0 only when x = 1.

Explanation

Any time the denominator is equal to 0, that value of x is not included in the functional value. So, x / (x^2-2x+1) = x / (x-1)^2. (x-1)^2 = 0 only when x = 1.

15. If a^2 = b^2, which of the following is/are always true? I. a = b II. |a| = |b| III. |a - b| = 0

I only

II only

I and II

I and III

I, II, and III

If a^2 = b^2, then a = + or - b.

So, I is not true because a could be = -b
II is true because |-b| = |b|
III is false because |b - (-b)| = |2b| = 2|b| is not equal to 0.

Explanation

If a^2 = b^2, then a = + or - b.

So, I is not true because a could be = -b
II is true because |-b| = |b|
III is false because |b - (-b)| = |2b| = 2|b| is not equal to 0.

16. A number is called "round" if it contains at least one zero as a digit. How many three-digit numbers are "round?"

153

171

178

179

215

An easy strategy here is to just count the number of "round" numbers between 100 and 199 and multiply that number by 9 for every hundreds digit (100, 200, 300... to 900). So, there are 19 "round" numbers through 100-199 (100, 101, 102 ... 109 = 10 + and 110, 120, 130... 190 = 9). 19 * 9 = 171

Explanation

An easy strategy here is to just count the number of "round" numbers between 100 and 199 and multiply that number by 9 for every hundreds digit (100, 200, 300... to 900). So, there are 19 "round" numbers through 100-199 (100, 101, 102 ... 109 = 10 + and 110, 120, 130... 190 = 9). 19 * 9 = 171

17. If m & n = (m + n)^(m - n), what is 2 & (2 & 2)?

2

3

4

6

8

First, do what is in the ( )'s. (2 & 2) = (2 + 2) ^ ( 2 - 2) = 4^0 = 1.

Next, plug in 1 for (2 & 2) to get 2 & 1 = (2 + 1)^(2 - 1) = 3^1 = 3.

Explanation

First, do what is in the ( )'s. (2 & 2) = (2 + 2) ^ ( 2 - 2) = 4^0 = 1.

Next, plug in 1 for (2 & 2) to get 2 & 1 = (2 + 1)^(2 - 1) = 3^1 = 3.

18. A three-digit number is called "big" if any two of its digits are equal. How many three-digit numbers are "big?"

112

146

214

252

316

Start with 100-199 and then multiply the result by 9 for each hundreds digit. 100, 101, 110, 111, 112, 113, ... 119, 121, 122, 131, 133, 141, 144, 151, 155... = 2 + 10 + 2(8) = 12 + 16 = 28. 28 * 9 = 252

Explanation

Start with 100-199 and then multiply the result by 9 for each hundreds digit. 100, 101, 110, 111, 112, 113, ... 119, 121, 122, 131, 133, 141, 144, 151, 155... = 2 + 10 + 2(8) = 12 + 16 = 28. 28 * 9 = 252

19. Which of the following statements is always true?

|(a+b)^2| < |a^2| + |b^2|

|a^2 + b^2| > (a+b)^2

|a^2 + b^2| >= |a+b|^2

|a^2 + b^2 - 1|

|a + b^2| > |a^2 - b|

Since |X + Y| is always less than or equal to |X| + |Y|, substitute a^2 for X and b^2 for Y to find |a^2 + b^2| is less than or equal to |a^2| + |b^2|

Explanation

Since |X + Y| is always less than or equal to |X| + |Y|, substitute a^2 for X and b^2 for Y to find |a^2 + b^2| is less than or equal to |a^2| + |b^2|

20. The number 100 has two trailing zeros. How many trailing zeros does 100! have?

12

15

18

24

28

Every trailing zero indicates a factor of 10 or 5 * 2. Since there are many more factors of 2 than 5's, you should count the number of factors of 5 there are in 100! Since it is one big product (100 * 99 * 98...), count: 5, 10, 15, 20, 25 (5 * 5), 30, 35, 40, 45, 50 (5*5), 55, 60, 65, 70, 75 (5*5), 80, 85, 90, 95, 100 (5*5). The total number of 5's is 24.

Explanation

Every trailing zero indicates a factor of 10 or 5 * 2. Since there are many more factors of 2 than 5's, you should count the number of factors of 5 there are in 100! Since it is one big product (100 * 99 * 98...), count: 5, 10, 15, 20, 25 (5 * 5), 30, 35, 40, 45, 50 (5*5), 55, 60, 65, 70, 75 (5*5), 80, 85, 90, 95, 100 (5*5). The total number of 5's is 24.

21. If |a| < |b|, and a > b, which of the following is necessarily true?

|a + b| > |b| + |a|

|a + b| < a - b

|a| + |b| > 2|b|

|a - b| > a + b

None of the above

None of the given options are necessarily true under the conditions that the absolute value of a is less than the absolute value of b, and a is greater than b. Each statement can be disproven with counterexamples. These conditions do not support the conclusions drawn in the statements about absolute values and arithmetic operations.

Explanation

None of the given options are necessarily true under the conditions that the absolute value of a is less than the absolute value of b, and a is greater than b. Each statement can be disproven with counterexamples. These conditions do not support the conclusions drawn in the statements about absolute values and arithmetic operations.

22. If a two-sided coin is flipped three times, what is the probability that at least one head will show up?

1/8

3/8

1/2

2/3

7/8

The probability of one head showing up is equal to 1 - P(All tails), and the probability of all tails is (1/2)^3 = 1/8, so 1- 1/8 = 7/8

Explanation

The probability of one head showing up is equal to 1 - P(All tails), and the probability of all tails is (1/2)^3 = 1/8, so 1- 1/8 = 7/8

23. How many ways can Pete, Mary, Sue, and Joe stand in a line if Joe and Sue cannot stand next to each other?

4

6

12

16

18

Step 1: Total Arrangements

There are 4 people, so without any restrictions, they can be arranged in 4 factorial (4 x 3 x 2 x 1) ways, which equals 24 ways.

Step 2: Arrangements Where Joe and Sue Are Together

Treat Joe and Sue as one unit. This changes the problem to arranging 3 units: {Joe-Sue}, Pete, and Mary.

These 3 units can be arranged in 3 factorial (3 x 2 x 1) ways, which equals 6 ways.

Joe and Sue can switch places within their unit, and this can happen in 2 factorial (2 x 1) ways, which equals 2 ways.

Therefore, Joe and Sue can be together in 6 x 2 = 12 ways.

Step 3: Valid Arrangements (Joe and Sue Not Together)

Subtract the number of ways Joe and Sue are together from the total arrangements: 24 (total ways) - 12 (Joe-Sue together) = 12 ways

Explanation

Step 1: Total Arrangements

There are 4 people, so without any restrictions, they can be arranged in 4 factorial (4 x 3 x 2 x 1) ways, which equals 24 ways.

Step 2: Arrangements Where Joe and Sue Are Together

Treat Joe and Sue as one unit. This changes the problem to arranging 3 units: {Joe-Sue}, Pete, and Mary.

These 3 units can be arranged in 3 factorial (3 x 2 x 1) ways, which equals 6 ways.

Joe and Sue can switch places within their unit, and this can happen in 2 factorial (2 x 1) ways, which equals 2 ways.

Therefore, Joe and Sue can be together in 6 x 2 = 12 ways.

Step 3: Valid Arrangements (Joe and Sue Not Together)

Subtract the number of ways Joe and Sue are together from the total arrangements: 24 (total ways) - 12 (Joe-Sue together) = 12 ways

24. A rectangle's length is twice its width. If the perimeter of the rectangle is 30 meters, what is the area of the rectangle?

20 square meters

36 square meters

40 square meters

50 square meters

First, let's define the width of the rectangle as w meters. According to the problem, the length (l) is twice the width, so we can express this as: l = 2w

The formula for the perimeter (P) of a rectangle is given by: P = 2l + 2w Substituting the given perimeter and the expression for l, we get: 30 = 2(2w) + 2w 30 = 4w + 2w 30 = 6w w = 5 meters

Now that we know the width, we can find the length: l = 2w = 2*5 = 10 meters

The area (A) of the rectangle is calculated by multiplying the length and width: A = l * w = 10 * 5 = 50 square meters

Explanation

First, let's define the width of the rectangle as w meters. According to the problem, the length (l) is twice the width, so we can express this as: l = 2w

The formula for the perimeter (P) of a rectangle is given by: P = 2l + 2w Substituting the given perimeter and the expression for l, we get: 30 = 2(2w) + 2w 30 = 4w + 2w 30 = 6w w = 5 meters

Now that we know the width, we can find the length: l = 2w = 2*5 = 10 meters

The area (A) of the rectangle is calculated by multiplying the length and width: A = l * w = 10 * 5 = 50 square meters