ACT MATH PRACTICE TEST ONLINE 1

Question 1 :

If x > 0 and  x² -2x - 35 = 0, then the value of x² is 

(a) 16     (b) 25     (c) 36     (d) 49     (e) 64

Solution :

x² - 2x - 35 = 0

x² - 7x + 5x - 35 = 0

x(x - 7) + 5(x - 7) = 0

(x + 5)(x - 7) = 0

Equating each factor to 0, we get

x + 5 = 0 and x - 7 = 0

x = -5 and x = 7

Since x > 0, then the possible value of x is 7.

x² = 7² ==> 49

So, option d is correct.

Question 2 :

On a sale, a particular product is discounted by 20%. The original price was $100. Then the price is raised to $120. What is the percentage increase from the discounted price to the new price?

(a) 20     (b) 30     (c) 40     (d) 50     (e) 60

Solution :

Original price = $100

Discount = 20%

New price = $120

Price after discount = 80% of 100

= 0.80(100)

= 80

Percentage increase = (120 - 80)/80 x 100%

= (40/80) x 100%

= (1/2) x 100%

= 50%

50% increase, then option d is correct.

Question 3 :

If the radius of the circle is decreased by 20%, by what percent is its area decreased ?

(a) 26     (b) 36     (c) 46     (d) 56     (e) 66

Solution :

Let r be the radius, decreasing 20%, we get 80% of r.

Area of circle = πr²

After decreasing 20%

= π(0.80r)²

= π(0.64r²)

= 0.64πr²

100 - 64 ==> 36

So, option b is correct.

Question 4 :

In the triangle given below, AB = 18, DE = 12 and AD = 6. And also AB and DE are parallel, what is the length of DC?

(a) 18     (b) 11     (c) 16     (d) 12     (e) 14

Solution :

Since AB and DE are parallel, then in triangles CDE and CAB

∠CDE = ∠CAB

∠DAE = ∠ACB

The triangles CDE and CAB are similar, then the corresponding sides will be in the same ratio.

CE/BC = DE/AB = CD/AC

12/18 = CD/6

CD = (12/18) · 6

= 4

So, option e is correct.

Question 5 :

Find x, if x√x = (x√x)^x.

(a) -1     (b) 0     (c) 1     (d) 2     (e) -2

Solution :

x√x = (x√x)^x

(x√x)¹ = (x√x)^x

Since the bases are equal, the powers will be equal.

So, the value of x is 1.

Question 6 :

One yard is three feet and one foot is twelve inches. How many inches are 5 yards?

(a) 180     (b) 110     (c) 160     (d) 120     (e) 140

Solution :

1 yard = 3 feet

1 foot = 12 inches

5 yards = ? inches

5 yards = 15 feet

15 feet = 15 x 12

= 180

So, option a is correct.

Question 7 :

If two primes are multiplied, the result must be,

(a) prime   (b) odd   (c) even   (d) negative   (e) composite

Solution :

Considering two primes, 3 and 5.

The product of these two primes = 15. Then it is odd.

Question 8 :

What is the value of log ₉ √3?

(a) 0.4     (b) 1.0     (c) 0.125     (d) 0.5     (e) 0.25

Solution :

= log ₉ √3

= log ₉ (3)^1/2

= (1/2) log ₉ (3)

= (1/2)(1/log₃ 9)

= (1/2) (1/log₃ 3²)

= (1/2) (1/2log₃ 3)

= 1/2 (1/2)

= 1/4

= 0.25

So, option e is correct.

Question 9 :

If t < 0 and (t - 1)² = 16, what is the value of t² ?

(a) 4     (b) 16     (c) -4     (d) 9     (e) 36

Solution :

(t - 1)² = 16

t - 1 = √16

t - 1 = -4 and t - 1 = 4

t = -4 + 1 and t = 4 + 1

t = -3 and t = 5

Accordingly the condition, t < 0

t = -3

t² = (-3)² ==> 9

So, option d is correct.

Question 10 :

David is hired for a job that pays $600 per month and receives  10 % raise in each following month. In the third month, how much will David earn ?

(a) $600     (b) $660     (c) $726     (d) $756     (e) $776

Solution :

Original salary = $600

10% raise, then 110% of 600

= 1.10(600)

= 660

So, option b is correct.

Question 11 :

Carmen is playing with blocks. She arranges stacks of blocks so that each successive level of blocks has 1 fewer block than the level below it and the top level has 1 block. Such a stack with 3 levels is shown below. Carmen wants to make such a stack with 12 levels. How many blocks should she use to build this stack ?

a) 66    b) 78    c) 132    d) 144    e) 156

Solution :

= 1 + 2 + 3 + ............ + 12

Sum of the arithmetic sequence,

sn = (n/2)(a + l)

Here n = 12, a = 1 and l = 12

= (12/2) (1 + 12)

= 6(13)

= 78

So, the total number of blocks is 78.

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