# ASVAB Mathematics Knowledge Test 1

#### The volume of this box is

The volume of a cubic rectangle is length x width x height: 3 x 4 x 5 = 60

#### If n is a positive integer divisible by 7, and if n < 70, what is the greatest possible value of n?

If you know one multiple of a number, you can find the other multiples by adding and subtracting that number. The number 70 is divisible by 7. The greatest multiple that is less than 70 will be 70 − 7 = 63.

#### Blair’s average (arithmetic mean) on two physics tests is 7. What should Blair’s score be on the next test to have an overall average of 8 for all the tests?

The arithmetic mean is defined as the sum of the terms divided by the number of terms. 7 = sum / 2 14 = sum Adding the next test’s score will change the average to an 8, and the total terms to 3. 8 = (14 + new score) / 3 24 = 14 + new score 10 = new score

#### (4x + 2) (x + 1) =

This is a multiplying binomials problem. Multiply the first term in the first binomial by both terms in the second binomial: 4x^{2} + 4x
Then multiply the second term in the first binomial by each term in the second binomial:
2x + 2
Adding them all together you get:
4x^{2} + 4x + 2x + 2
4x^{2} + 6x + 2

#### 2 − 8 ÷ (2^{4} ÷ 2) =

Remember your order of operations:

- Complete operations within parentheses first.
- Calculate exponents.
- Then multiply and divide in order from left to right.
- Then add and subtract in order from left to right.

^{4}÷ 2) =

2 − 8 ÷ (16 ÷ 2) =

2 − 8 ÷ 8 =

2 − 1 =

1

#### 3(x − 4) = 18, what is the value of x?

Begin by dividing both sides by 3 to open the parenthesis: x − 4 = 6 Add 4 to both sides to find x: x = 10

#### A scoutmaster is preparing his troop for their annual fishing expedition. He surveys each of his scouts to find out how many fishing poles they own. What is the mode of this data set?

The mode of the data is the number that appears the most often. In this case the largest number of scouts (7) own 0 fishing poles.

#### Line l passes through the point (−1, 2). Which of the following CANNOT be the equation of line l?

The slope-intercept form of a line is y = mx + b. Since the line passes through (−1,2), there are three possibilities: the line will have a slope (the “m” in front of the “x” variable), it will be vertical (x = −1), or it will be horizontal (y = 2). Plug x = −1 into all four equations to see which equation is not satisfied. The only answer choice that doesn’t give us y = 2 is (B)

#### If 5^{11} = 5^{2} x 5^{m}, what is the value of *m*?

When multiplying two powers with the same base, you can always add the exponents. This means that 11 = 2 + m, so m = 9.

#### Which of these expressions is equivalent to: 3x^{3}y^{5} + 3x^{5}y^{3} − (4x^{5}y^{3} − 3x^{3}y^{5})

The key to this problem is distributing the negative sign. When distributing a negative sign, each term has a change of sign from negative to positive or from positive to negative. 3x^{3}y^{5} + 3x^{5}y^{3} − (4x^{5}y^{3} − 3x^{3}y^{5})

3x^{3}y^{5} + 3x^{5}y^{3} − 4x^{5}y^{3} + 3x^{3}y^{5}

6x^{3}y^{5} − x^{5}y^{3}

#### A rectangle is cut in half to create two squares that each have an area of 25. What is the perimeter of the original rectangle?

The formula to find the area of a square is side x side. If each square has an area of 25, then the dimensions of the rectangle are 5 x 10. The perimeter is the sum of all the sides, and would therefore be 5 + 10 + 5 + 10 = 30.

#### If XY = YZ, and angle Y measures 90 degrees in the figure above, which of the following CANNOT be concluded?

Here we are looking for the choice that is NOT true. Since the triangle is isosceles (XY = YZ), then a = b = 45. This means that a + b = 90. (A) and (C) must be true. For all right triangles, the Pythagorean Theorem applies, so (D) is also true. (B) is the correct answer because it is not true that XZ < XY. In a right triangle, the hypotenuse is always the longest side.

#### What is the value of this expression:

You need to add the 25 and 144 first, and then take the square root. The square root of 169 is 13.

#### If (3.2 + 3.3 + 3.5)w = w, then what is the value of w?

First add up the three numbers that are in parentheses: 10w = w 10w = w will only be true when w = 0.

#### There are two pizza ovens in a restaurant. Oven 1 burns three times as many pizzas as oven 2. If the restaurant had a total of 12 burnt pizzas on Saturday, how many pizzas did oven 2 burn?

Begin by setting up a system of equations to represent the situation, letting x represent oven 1 and y represent oven 2. x = 3y and x + y = 12 Substituting the first equation into the second: 3y + y = 12 so y = 3.

#### What is the area of this trapezoid?

The trapezoid consists of two smaller shapes: a triangle and a rectangle. Draw a line from point A across this figure to form the two shapes. Find the area of each shape and then add them together to find the area of the trapezoid.

Area of rectangle = length x width.

Therefore, the area of the rectangle is 18x20 = 360

Area of a triangle = 1/2(base)(height).

The base of the triangle is 20.

The height of the triangle is unknown.

Use the Pythagorean Theorem to find the triangle's height.
c^{2} = a^{2} + b^{2} where c is the triangle's hypotenuse.

29^{2} = 20^{2} + (height)^{2}

841 = 400 + (height)^{2}

441 = (height)^{2}

The square root of 441 is 21

height = 21.

Therefore, area of the triangle = (1/2)(20)(21) = 210. The total area of the trapezoid = 360+210 = 570.

#### Given a circle with a radius of 0.85 meters, find the area of the circle.

The area of a circle is equal to π r^2. Putting in the variables we know, we find the area of the circle to be 2.27 pi.

#### Calculate the area of a parallelogram with a base of 3 feet and a height of 1.2 feet.

The area of a parallelogram is A = bh, and the area is 1.2 x 3, which is 3.6 feet squared.

#### A given rectangle has an area of 24 square feet. Which of the following cannot be one of its side lengths?

Without knowing either side length, there are infinite possibilities of side lengths for a rectangle of area 24 square feet. For example, you could have one side of 480 feet, and one side be 0.05 feet, and still get an area of 24 square feet.

#### The base of a right triangle is 1 foot, and the interior angles are 45-45-90. What is its area?

If you draw out this triangle, you'll see that is a right isosceles triangle with two equal sides of 1 foot. This means that the base is 1 foot and the height is 1 foot. Using the formula for area of triangles, A = 1/2bh, we find the area is 0.5 feet squared.

#### The circumference of a circle is 20 cm. What is the approximate radius of the circle?

The formula for the circumference of a circle is C = 2πr. Plug in the given circumference and solve for the radius. C = 2πr 20= 2πr 10 = πr Use 3.14 as an approximate value for π and divide to solve. Round to the nearest tenth. r = 3.2

#### Calculate the area of the circle shown in the figure.

The area of a circle is equal to πr^{2}, and with a radius of 10 cm, the area is equal to 100 square centimeters multiplied by pi.

#### In the figure below, line A is parallel to line B. What is the value of angle x?

First, we find the supplementary angle to 40 degrees, which is 140 degrees. That tells us the angle opposite is 140 degrees. Because they are two parallel lines, then the mirrored angle must also be 140 degrees.

#### What is the distance between the points (2,4) and (-1, 8)?

To find the distance, we use pythagoreas theorem. First calculate the distances in terms of x and y. The x difference is 3 and the y difference is 4. Then, √(3^{2} + 4^{2}) = distance, which is 5.

#### Which of the following statements about trapezoids is true?

Trapezoids do not need to have any angles that are equal to one another. If the two bases of a trapezoid are parallel, then it is possible to form a trapezoid with 2 triangles and a rectangle, but never 3 triangles.

#### Two parallel lines are intersected by a third line, which is perpendicular to the first parallel line. The value of another angle made by the intersecting lines is:

If the first line is intersected and forms a 90 degree angle, then that means that all other angles are also equal to 90 degrees.

#### Given that x = 0.5 and y = 4, what is the value of 4x^{2}(2y+2)?

The value of x squared is 0.25, and as a result 4x^{2} is equal to 1. This means the equation simplifies to 1(2y+2). Since y has a value of 4, if we substitute and solve we find the answer is 10.

#### You are told that one of the internal angles of a parallelogram is 54 degrees. This means at least one other angle must be equal to:

A parallelogram has 2 sets of angles that are equal to one another. That means if one angle is 54 degrees, at least one other angle (the opposite angle) must also be 54 degrees. The other two angles are both 126 degrees.

#### The area of a circle is given as 24π. What is its radius?

The formula for the area of a circle is A = πr^{2}. Plug in the area you are given and solve for the radius.
A = πr^{2}
24π = πr^{2}
24 = r^{2}
r = √24 = √4 √6=2√6

#### A medium pizza has a diameter of 14 inches. What is its circumference?

The circumference of a circle is equal to the diameter multiplied by π.

#### What is one possible value of x in this equation? x^{2}-3x-18=0

Factor the quadratic using the reverse-FOIL method to solve. x2-3x-18=0 (x+3)(x-6)= 0 x={-3,6} -3 isn’t one of the answer choices, so the correct answer must be 6.

#### A line segment is always constrained between:

A line segment is specifically constrained by 2 points. If not constrained, it would just be a line, and lines go on in either direction for infinity.

#### The equation of a line is given as: y = 3x + 5. Which of the following points does not lie on the line?

Points that lie on the line with the equation given must correctly fill in the equation. For example, answer A gives an X value of 1, from which we can calculate y = 3(1) +5 = 8. Thus, the point (1, 8) must lie on the line. Answer C does not lie on the line, because if we enter an X value of 3, then y = 3(3) + 5 = 14, which does not match with the point’s value of (3,16). Thus, Answer C does not lie on the line.

#### A square has one side with length 4.5 feet. The area of the square is:

Squares have 4 sides that are the same length, and the area of a square is equal to its side squared, in this case 4.5 x 4.5, which is equal to 20.25.

#### In the circle seen below, what is the value of x?

Given any chord, if a second chord is perpendicular to that chord and also passes through the center of the circle, then the first chord must be bisected. This means the first chord has been divided into two equal halves, meaning x must also be 8.

#### What angle is complementary to 47 degrees?

Two angles are coplementary if they add up to 90 degrees. This means that the complementary angle to 47 degrees is equal to 90-47 = 43 degrees.

#### If 4.5 < x ≤ 8.0, then x cannot be equal to:

< means less than. If 4.5 is less than x, it cannot be equal to x.

#### Find the slope of the line seen in the figure.

Slope is equal to rise over run (y/x). In the figure, we can see that as x increases by 3, y increases by 2. Thus, the slope is 2/3, or 0.66.

#### How long is the line segment shown on the number line below?

From the zero, the line is 7 units long on the positive side and 2 units long on the negative side. This adds up to a total of 9 units in length.

#### What is the value of angle x in the figure below?

The two chords crossing each other have equal opposite angles, with the smaller angle being supplementary to the larger one. Thus, the supplementary angle is 50 degrees, and since we know that is is a right triangle, that means the remaining angle is 40 degrees.

#### 15y -20 = 110. y = ?

First, we add 20 to both sides to arrive at 15y = 130. Then, we divide both sides by 15 to arrive at y = 130/15, which is equal to 8.66.

#### Which of the following is not a type of quadrilateral?

A quadrilateral is defined as a shape that has 4 sides. Squares, trapezoids, and parallelograms all have 4 sides. Pentagons have 5 sides.

#### What inequality is represented by the number line seen in the figure?

The line indicates values that are greater than 3. An open circle around the 3 indicates that 3 itself is not included, so the equation is x > 3. If the circle had been filled in, 3 would be included and the equation would be x > 3 (x is greater than or equal to 3).

#### Solve for the value of y in the following system: 2x - 4y = 6 5x + 2y = 9

You can solve a system of equations using either combination or substitution. With combination, multiply one of the equations by a factor that would make one of the variables cancel out. In this case, you could multiply 5x + 2y = 9 by 2 so that the y terms will cancel out. 5x + 2y = 9 2(5x + 2y = 9) 10x + 4y = 18 Now line this equation up with the other one and add the like terms. 10x + 4y = 18 +(2x - 4y = 6 ) 12x = 24 x = 22 Now that you’ve solved for x, you can plug 2 into either original equation to get the value of y. 5x + 2y = 9 5(2) + 2y = 9 10 + 2y = 9 2y = -1 y = -1/2 You can also solve by substitution. Solve one of the equations for x and then plug that expression into the other equation to solve for y. 2x – 4y = 6 2x = 4y + 6 x = 2y + 3 Plug 2y+ 3 in for x in the other equation to solve for y. 5x + 2y = 9 5(2y + 3) + 2y = 9 10y + 15 + 2y = 9 12y + 15 = 9 12y = -6 y = -1/2

#### If two angles in a triangle measure 35 degrees and 47 degrees, what is the value of the third angle?

The interior angles of a triangle must add up to 180 degrees. 180 – 35 – 47 = 98 degrees. Thus, the last angle in the triangle must be 98 degrees.

#### What is the perimeter of the shape seen in the figure?

The angles in the figure tell you that the triangle must be equilateral (150-90 = 60 degrees for the corners of the triangle). That means all three sides of the triangle are 10 cm. This also means the base of the figure is 10 cm + 6 cm = 16 cm. the left side is 14 cm, adding up the values given on the right side. Thus, the total perimeter is 10 cm + 10 cm + 14 cm + 16 cm + 6 cm + 6 cm + 8 cm, which is 70 cm total.

#### What is the expanded equation from (3x+4)(x-2) = 0?

We have to use FOIL (First, outer, inner, last) to correctly expand the equation into 3x^{2} -6x +4x -8. After we add all similar terms together, we arrive at the answer 3x^{2} -2x -8.

#### The surface area of a rectangular prism with sides 3 x 5 x 5 inches is:

The area of each of the three unique sides are 15, 15, and 25. We need to double this area because each side is mirrored in a rectangular solid. This gives us a total surface area of 110 square inches.

#### Calculate the area of the trapezoid in the figure in square feet.

The area of a trapezoid is equal to 1/2(b1+b2)h. For this trapezoid, that is 1/2(2 + 4.5)(2) = 6.5 square feet.

#### Given 4x = 15, x = ?

We divide both sides of the equation by 4 to find that x = 15/4, or 3.75

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