A square field has an area of 22,500 square feet. To the nearest foot, what is the diagonal distance across the field?

150 feet
178 feet
191 feet
212 feet
260 feet

Solution

The area of the field is 22,500 square feet, so plug this value into the formula for the area of a square to find the length of the side:

A = S²

22,500 = S²

150 = S

So the side of the field measures 150 feet. The diagonal of a square is the hypotenuse of a 45-45-90 triangle, which has three sides in a ratio of x : x : x√2.

Thus, the length of this diagonal is 150√2 ≈ 212.(You can also use the Pythagorean theorem to find this result.)

Which of the following is equal to (n– 3)² if n = 11?

(n + 6)(n – 6)
(n + 5)(n – 7)
(n + 4)(n – 8)
(n + 3)(n – 9)
(n + 2)(n – 10)

Solution

If n = 11, then:

(n – 3)² = (11 – 3)² = 8² = 64

Substitute 11 for n into each answer until you find one that equals 64:

(n + 6)(n – 6) = (17)(5) = 85 Wrong!

(n + 5)(n – 7) = (16)(4) = 64 You’ve found a match!

In the following figure, the two triangles are similar. The lengths of their sides are shown. What is the value of yin terms of x?

x + 1
x + 2
x + 3
x + 4
x + 5

Solution

The small triangle has a side with the length of 3, and the large triangle has a corresponding side with the length of 6, which is twice as long.

The triangles are similar, so each side of the large triangle is twice the length of the corresponding side of the small triangle.

One side of the small triangle has a length of 2, so the corresponding side of the large triangle is 4; thus, x = 4.

One side of the small triangle has a length of 4, so the corresponding side of the large triangle is 8; thus, y = 8.

Therefore, y = x + 4.

Which of the following inequalities is represented by the following number line?

–5 < x < 5
–5 ≤ x < 5
–5 < x ≤ 5
|x| < 5
5 ≤ |x|

Solution

The number line includes all values that are both greater than or equal –5, but strictly less than 5, so the correct answer is Choice (B).

When you multiply a number by 4 and then subtract 7, the result is the same as if you first subtracted 7 from the same number and then multiplied by 11. What is the number?

Solution

Let x equal the number and then change the words into an equation and solve for x:

4x – 7 = 11(x – 7)

4x – 7 = 11x – 77

70 = 7x

10 = x

William earned $3,200 per month as a teacher for the ten months from September to June. Then he took a job as a barista at a local café, where he earned $2,000 per month during July and August. What was his average monthly pay for the 12 months?

$2,400
$2,500
$2,600
$2,800
$3,000

Solution

William earned $3,200 ×10 = $32,000 as a teacher, and he earned $2,000 × 2 = $4,000 as a barista.

So he earned $32,000 + $4,000 = $36,000 over the 12 months.

Plug these values into the formula for the mean:

$Mean \;\; = \frac{Sum \;\; of \;\; values}{Number \;\; of \;\; values} = \frac{36,000}{12} = 3,000$

The number 4 is the smallest positive integer that has exactly three factors: 1, 2, and 4. If k is the next-highest integer that also has exactly three factors, what is the sum of the three factors of k?

Solution

The greatest of the five possible answers is 20, so k is less than 20. Use trial and error to find the value of k:

Factors of 5 : 1 5

Factors of 6 : 1 2 3 6

Factors of 7 : 1 7

Factors of 8 : 1 2 4 8

Factors of 9 : 1 3 9

Thus, k = 9, and the sum of the factors of k is 1 + 3 + 9 = 13.

If 5x + 3 = 10x – 17, what is the value of x?

Solution

Solve for x:

5x + 3 = 10x - 17

20 = 5x

x = 4

Beth got a job painting dorm rooms at her college. At top speed, she could paint 5 identical rooms during one 6-hour shift. How long did she take to paint each room?

50 minutes
1 hour and 10 minutes
1 hour and 12 minutes
1 hour and 15 minutes

Solution

Six hours equals 360 minutes, so Beth paints 5 rooms in 360 minutes.

She paints 1 room in 360 ÷ 5 = 72 minutes, which equals 1 hour and 12 minutes.

Which of the following numbers completes the sequence 3, 8, 14, 21, 29, ___?

Solution

The numbers increase at a somewhat steady rate, so you have to figure out how much you have to add to each number to produce the next in the sequence:

3 + 5 = 8;

8 + 6 = 14;

14 + 7 = 21;

21 + 8 = 29, and so on.

The rule for the sequence is to add successively larger numbers to each number; therefore, the next number is 29 + 9 = 38.

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