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 :

g(x) = |7x/9 – 40|

The function g is defined by the given equation. For which of the following values of a does g(a) = a?

A) –180

B) 27/2

C) 45/2

D) 360/7

Solution :

Problem 5 :

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

Solution :

Problem 6 :

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 7 :

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 8 :

–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 9 :

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 10 :

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 11 :

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 12 :

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 13 :

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

= 108% ⋅ 0.8x

= 1.08 ⋅ 0.8x

Given : The total amount Alma paid to the cashier was p dollars, including an 8 percent sales tax on the discounted price.

Then, we have

1.08 ⋅ 0.8x = p

x = p/[(0.8)(1.08)]

The correct answer choice is (D).

Problem 14 :

If a = 5√2 and 2a = √(2x), what is the value of x?

Solution :

2a = √(2x)

Substitute a = 5√2.

2(5√2) = √(2x)

10√2 = √(2x)

Square both sides.

[10√2]² = [√(2x)]²

10²[√2]² = 2x

100(2) = 2x

200 = 2x

Divide both sides by 2.

100 = x

Problem 15 :

Which of the following is the value of the above expression?

A) -1

B) 0

C) 1

D) 10

Solution :

Problem 16 :

The function f is defined by f(x) = a√(x + b), where a and b are constants and a ≠ 0. In the xy-plane, the graph of y = f(x) passes through the point (–24, 0), and f(24) is less than 0. Which of the following must be true?

A) a > 0

B) b < 0

C) a > b

D) a < b

Solution :

Problem 17 :

Which of the following is equivalent to the expression above?

Solution :

Problem 18 :

If x + y = 13, then what is the value of

(x – 5)¹⁰⁰ – (y – 8)¹⁰⁰?

Solution :

Problem 19 :

nx - 9k = -2x + 3n

In the equation above, n and k are constants. If the equation has no solution, what cannot be the value of n + k?

Solution :

Problem 20 :

The table gives the volume of two similar cylinders. If the radius of cylinder A is 9 cm, what is the value of s – t?

Volume = πr²h

Surface Area = 2πr² + 2πrh

Solution :

Problem 21 :

Find in what ratio will the total wages of the workers of a factory be increased or decreased, if there is a decrease in the number of workers in the ratio 15 : 11 and an increase in their wages in the ratio 22 : 25.

A) Increase, 5 : 6

B) Decrease, 6 : 5

C) Increase, 3 : 4

D) Decrease, 4 : 3

Solution :

Problem 22 :

The value of the expression above can be written in the form a +√b. What is the value of a + b?

Solution :

Problem 23 :

In the figure above, what is the area of the square (shaded region)?

Solution :

Problem 24 :

In the given system of equations, k is a constant. If the system has no solution, what is the value of k?

Solution :

Problem 25 :

In the xy-plane, the graph of the given equation is a circle. Which point lies on this circle?

(x + 7)² + (y - 11)² = 25

A) (-7, 11)

B) (11, -7)

C) (√11 + 7, √14 - 11)

D) (√11 - 7, √14 + 11)

Solution :

Problem 26 :

The area of a triangle is equal to x² square inches. The base of the triangle is 5 + 2x inches, and the height of the triangle is x − 2 inches. What is the value of x?

A) 2.5

B) 2.7

C) 5

D) 10

Solution :