GRADE 9 MATH PRACTICE PROBLEMS

Problem 1 :

Which of the following is an equivalent form of the expression

15 x + 24 ax ?

A) 39ax²  B) 39 (a + 2x)   C) (5 + 8a)x   D) (15 + 24 a) x

Solution :

= 15 x + 24 ax

Factoring 3x, we get

= 3x(5 + 8a)

Factoring x alone,

= x(15 + 24a)

So, option D is correct.

Problem 2 :

The formula d = rt is used to calculate the distance an object travels over a period of time, at a constant rate, r. Based on this formula, what is the rate, r, in terms of d and t ?

A) r = d/t   B) r = dt     C) r = t/d    D) r = d - t

Solution :

d = rt

To solve for r, we have to divide by t on both sides.

d/t = r

r = d/t

Problem 3 :

Which of the following ordered pairs (x , y) satisfies both equations y = x² + 3 x − 4 and x = y − 4 ?

A) (0, −4)     B) (2, 6)     C) (3, 14)    D) (5, 9)

Solution :

y = x² + 3 x −4 -----(1)

x = y − 4 -----(2)

Applying the value of x in (1), we get

y = (y - 4)² + 3 (y - 4) −4

y = y² - 8y + 16 + 3y - 12 - 4

y = y² - 5y

y² - 5y - y = 0

y² - 6y = 0

y(y - 6) = 0

y = 0 and y = 6

So, the required points are (-4, 0) and (2, 6). Option B is correct.

Problem 4 :

Which of the following is a solution to the equation

2x² + 4x = 3 + 3x² ?

A) −1    B) 0     C) 3     D) 6

Solution :

2x² + 4x = 3 + 3x²

0 = 3x² - 2x² - 4x + 3

 x² - 4x + 3 = 0

(x - 1)(x - 3) = 0

x = 1 and x = 3

So, option C is correct.

Problem 5 :

−3x − 4 y = 20

x − 10 y = 16

If (x, y) is the solution to the system of equations above, what is the value of x ?

A) −14     B) −12    C) −4     D) 16

Solution :

−3x − 4 y = 20 -----(1)

x − 10 y = 16 -----(2)

x = 16 + 10y

Applying the value of x in (1), we get

-3(16 + 10y) - 4y = 20

-48 - 30y - 4y = 20

-48 - 34y = 20

-34y = 20 + 48

-34y = 68

y = 68/(-34)

y = -2

Applying the value of y, we get

x = 16 + 10(-2)

x = 16 - 20

x = -4

So, the value of x is -4, option C is correct.

Problem 6 :

The equation y = 36 + 18x models the relationship between the height y, in inches, of a typical golden delicious apple tree and the number of years, x, after it was planted. If the equation is graphed in the xy-plane, what is indicated by the y-intercept of the graph?

A) The age, in years, of a typical apple tree when it is planted

B) The height, in inches, of a typical apple tree when it is planted

C) The number of years it takes a typical apple tree to grow

D) The number of inches a typical apple tree grows each year

Solution :

y = 36 + 18x

Comparing with y = mx + b

y = 18x + 36

b = 36

It represents the height, in inches, of a typical apple tree when it is planted.

Problem 7 :

Giovanni wants to buy shirts that cost $19.40 each and sweaters that cost $24.80 each. An 8% sales tax will be applied to the entire purchase. If Giovanni buys 2 shirts, which equation relates the number of sweaters purchased, p, and the total cost in dollars, y ?

(A) 1.08 (38.80 + 24.80 p) = y

(B) 38.80 + 24.80 p = 0.92 y

(C) 38.80 + 24.80 p = 1.08 y

(D) 8 (0.92 38.80 + 24.80 p) = y

Solution :

Cost of one shirt = 19.40

Cos to 2 shirts = 2(19.40)

= 38.8

Number of sweaters purchased = p

Total cost = 108% of (cost of 2 shirts + cost of sweaters)

= 1.08(38.8 + 24.80p)

Required cost = 1.08(38.8 + 24.80p)

So, option A is correct.

Problem 8 :

A line is graphed in the xy-plane. If the line has a positive slope and a negative y-intercept, which of the following points cannot lie on the line?

A) (−3, −3)    B) (−3, 3)     C) (3, −3)    D) (3, 3)

Solution :

Slope is positive and y-intercept is negative.

When a line passes through (-3, -3), it may be raising line or falling line.

When we draw the raising line, we get positive slope and negative y-intercept.

Problem 9 :

A parachute design uses 18 separate pieces of rope. Each piece of rope must be at least 270 centimeters and no more than 280 centimeters long. What inequality represents all possible values of the total length of rope x, in centimeters, needed for the parachute?

A) 270 ≤ x ≤ 280      B) 4,860 ≤ x ≤ 4,870

C) 4,860 ≤ x ≤ 5,040      D) 5,030 ≤ x ≤ 5,040

Solution :

Minimum length of rope = 270 cm

Maximum length of rope = 280 cm

Total length of rope = x

270(18) ≤ x ≤ 280(18)

4860 ≤ x ≤ 5040

So, option C is correct.

Problem 10 :

A carpenter has $60 with which to buy supplies. The carpenter needs to buy both nails and screws. Nails cost $12.99 per box, and screws cost $14.99 per box. If n represents the number of boxes of nails and s represents the number of boxes of screws, which of the following systems of inequalities models this situation?

A) 1 2.99 n + 14.99 s ≥ 60, n +s ≤1

B) 1 2.99 n + 14.99 s ≤ 60, n +s ≤1

C) 12.99 n + 14.99 s ≥ 60, n ≥1 and s ≥1

D) 12.99 n + 14.99 s ≤ 60, n ≥1 and s ≥1

Solution :

Number of nails = n

Number of screws = s

Cost of nail per box = 12.99

Cost of screw = $14.99

12.99n + 14.99 x ≤ 60, n ≥1 and s ≥1

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