DIGITAL SAT MATH PROBLEMS AND SOLUTIONS
(Part - 1)

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Problem 1 :

The population of the town of Smithville doubled every 75 years from 1659 to 1959. The population of this town was 240,000 in 1959. What was the population of this town in 1659?

Solution :

Problem 2 :

Line h is passing through the points (2, 5) and (7, 40). Line k is the result of translating line h up 23 units. What is the x-intercept of the line k?

A) (9/7, 0)

B) (–9/7, 0)

C) (2, 0)

D) (–2, 0)

Solution :

Problem 3 :

If y/(x + y) =7/12, then what is the value of x/y?

A) 5/12

B) 5/7

C) 7/5

D) 7/19

Solution :

Problem 4 :

Which equation defines function g, if the graph of y = g(x) - 10 is shown above?

Solution :

Problem 5 :

If c is a constant in the equation 10x² + c = –5x, and the equation has no real solutions, what is the value of c ?

A) -20

B) -5

C) 0

D) 1

Solution :

Problem 6 :

3x – 4y = 17

In the xy-plane, the graph of a line with an x-intercept of (c, 0) and a y-intercept of (0, k), where c and k are constants, can be represented by the equation above. What is the value of c/k?

A) -4/3

B) -3/4

C) 3/4

D) 4/3

Solution :

Since (c, 0) is the x-intercept of the given line, it must through the point (c, 0). So, substitute x = c and y = 0 into the equation.

3c – 4(0) = 17

3c – 0 = 17

3c = 17

Divide both sides by 3.

c = 17/3

Since (0, k) is the y-intercept of the given line, it must through the point (0, k). So, substitute x = 0 and y = k into the equation.

3(0) – 4k = 17

–4k = 17

Divide both sides by -4.

k = -17/4

Find the value of c/k.

c/k = (17/3) ÷ (-17/4)

c/k = (17/3) x (-4/17)

c/k = -4/3

The correct answer choice is (A).

Problem 7 :

–7 + 2f = cg

21g + 21 = 6f – 15g

If c is a constant, and the system of equations shown above has infinitely many solutions, what is the value of c ?

Solution :

Problem 8 :

Triangle A has angles measuring 30º, 60º, and 90º. What
is the perimeter, in centimeters, of this triangle if the
smallest side has a length of 15 centimeters?

A) 15√3

B) 15 + 15√3

C) 45 + 15√3

D) 45√3

Solution :

Problem 9 :

x³ - 3x² + 3x - 9 = 0

For what real value of x is the equation above true?

Solution :

x³ - 3x² + 3x - 9 = 0

Factor by grouping.

x²(x- 3) + 3(x - 3) = 0

(x - 3)(x² + 3) = 0

x - 3 = 0 or x² + 3 = 0

x - 3 = 0

x = 3

(Real)

x² + 3 = 0

x² = -3

x = ±√-3

(Imaginary)

The given equation is true for x = 3.

Problem 10 :

A parabola represents the graph of the function f in the xy-plane, where y = f(x). If the vertex of the parabola is (5, –4) and one of the x-intercepts is (–1.5, 0), what is the other x-intercept?

A) (-6.5, 0)

B) (1.5, 0)

C) (3.5, 0)

D) (11.5, 0)

Solution :

(–1.5, 0) is one of the x-intercepts of the parabola. Let (a, 0) be the other x-intercept. The average of the x-coordinates of two x-intercepts of a parabola is equal to the x-coordinate of the vertex.

Given : Vertex = (5, -4).

Then, we have

(-1.5 + a)/2 = 5

Multiply both sides by 2.

-1.5 + a = 10

Add 1.5 to both sides.

a = 11.5

(a, 0) = (11.5, 0)

The other x-intercept is (11.5, 0).

The correct answer choice is (D).

Problem 11 :

If a + b = 8 and 27^a/3^b = 81, what is the value of a?

A) 3

B) 4

C) 5

D) 6

Solution :

27^a/3^b = 81

(3³)^a/3^b = 3⁴

3³^a/3^b = 3⁴

3³^{a - b} = 3⁴

In the equation above, two powers are equal with the same base. Then, the exponents must be equal.

3a - b = 4 ----(1)

Given : a + b = 8 ----(2).

Add (1) and (2).

4a = 12

Divide both sides by 4.

a = 3

The correct answer choice is (A).

Problem 12 :

Alma bought a laptop computer at a store that gave a 20 percent discount off its original price. The total amount she paid to the cashier was p dollars, including an 8 percent sales tax on the discounted price. Which of the following represents the original price of the computer in terms of p?

A) 0.88p

B) p/0.88

C) (0.8)(1.08)p

D) p/[(0.8)(1.08)]

Solution :

Let x be the original price of the computer.

Price of the computer after the 20% discount :

= (100 - 20)% ⋅ x

= 80% ⋅ x

= 0.8x

Price of the computer after 8% sales tax on the discounted price 0.8x :

= (100 + 8)% ⋅ 0.8x