# ASVAB Arithmetic Reasoning Test

#### With what number must 3.475817 be multiplied in order to obtain the number 34,758.17?

The decimal must be moved four places to the right. To do this, we must multiply by a number with four zeroes. The correct answer is 10,000.

#### How much greater is the value of 3x + 5 than the value of 3x − 7?

The easiest way to do this is to pick a number for x. Let’s say x = 3. 3(3) + 5 = 9 + 5 = 14 3(3) − 7 = 9 − 7 = 2 The correct answer is 14 − 2 = 12.

#### Which of the following is NOT a factor of 90?

A factor must divide evenly into its multiple. 12 cannot be a factor of 90 because 90 divided by 12 = 7.5.

#### Lisa and Robert have taken the same number of photos on their school trip. Lisa has taken 3 times as many photos as Claire and Robert has taken 12 more photos than Claire. How many photos has Claire taken?

Translate the information in the question from “English” to “Math.” L = R L = 3C R = C + 12 We can substitute R for L in the second equation: R = 3C. If R is equal to both 3C and C + 12, we can say 3C = C + 12, and solve for C. 3C = C + 12 2C = 12 C = 6

#### The sum of 7 numbers is greater than 140 and less than 210. Which of the following could be the average (arithmetic mean) of the numbers?

The formula for the average of a set of numbers is the sum of the numbers divided by the number of terms. Avg = 140 / 7 Avg = 20 Avg = 210 / 7 Avg = 30 Therefore, the sum must be between 20 and 30.

#### The hour hand of a watch rotates 30 degrees every hour. How many complete rotations does the hour hand make in 6 days?

There are 360 degrees in a complete circle, so 360/30 = 12 hours to make one full circle. In 6 days there are 24 hours x 6 = 144 hours total. The total number of rotations will be 144/12 = 12.

#### Which of the following expressions is equal to 3^{8} when *p* = 3^{5}?

3^{8} = 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3, or 3^{5} x 3^{3} = *p* x 3^{3}

*p* x 3^{3} = 27*p*

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#### If y + 3y + 5y = −18, then what is the value of y?

When terms with the same variable are in the same equation, they can be combined. y + 3y + 5y = 7y 9y = −18 y = −18/9 y = −2

#### If b does not equal zero, and ab = b/4, what is the value of a?

To solve for a, divide both sides of the equation by b: ab = b/4 (ab)/b = (b/4)/b a = (b/4)*1/b a = 1/4

#### If a store adds 50 chairs to its current inventory, the total number of chairs will be the same as three-halves the current inventory of chairs. If the manager wants to increase the current inventory by 40%, what will the new inventory of chairs be?

This word problem requires careful translation. Let’s say t = total current inventory of chairs. The first sentence states that 50 + t = (3/2)t. First solve for the current inventory: 50 + t = (3/2)t 50 = (3/2)t − t 50 = (1/2)t 100 = t The manager wants to increase this by 40%. 40% of 100 is 40, so the new inventory will be 140.

#### On a map, the length of the road from Town F to Town G is measured to be 18 inches. On this map, ¼ inch represents an actual distance of 10 miles. What is the actual distance, in miles, from Town F to Town G along this road?

Here we are given a ratio: ¼ inch on the map = 10 miles, so 1 inch on the map = 40 miles. If the map-distance between the towns is 18 inches, then the actual distance must be 18 x 40 = 720.

#### How many 1/3 pound paperback books together weigh 25 pounds?

If each book weighs 1/3 pound, then 1 pound = 3 books. We can set up a proportion to solve: 1 pound / 3 books = 25 pounds / x books. Now cross-multiply: (1)(x) = (3)(25) X = 75

#### The first four terms in a sequence are shown below. What is the sixth term in the sequence?

Begin by examining the sequence for a pattern. In order to go from 3 to 6, 3 must be added; moving from 6 to 11 requires 5 to be added; moving from 11 to 18 requires 7 to be added. The pattern emerges here — adding by consecutive odd integers. The 5th term is equal to 18 + 9 = 27, and the 6th term is equal to 27 + 11 = 38.

#### Each year, a cyber café charges its customers a base rate of $25, with an additional $0.30 per visit for the first 50 visits, and $0.10 for every visit after that. How much does the cyber café charge a customer for a year in which 72 visits are made?

Translate the information into arithmetic. The café charges $25 + $0.30(first 50) + $0.10(additional after 50). For 72 visits there are 50 visits with an additional 22 visits. $25 + $0.30(50) + $0.10(22) =$25 +$15 + $2.20 =$42.20

#### If Jill needed to buy 9 bottles of soda for a party in which 12 people attended, how many bottles of soda will she need to buy for a party in which 8 people are attending?

We can set up a proportion to solve: 9 bottles / 12 people = x bottles / 8 people. Cross-multiply to solve a proportion: (9)(8) = (12)(x) 72 = 12x 6 = x

#### Steve bought a total of 6 packages of pens, and each package contained either 3 or 7 pens. If exactly 4 of the packages Steve bought contained 7 pens, how many pens did Steve buy?

If Steve bought 4 packages of 7 pens and 6 packages total, then he must have purchased 2 packages of 3 pens. 4(7) + 2(3) = 28 + 6 = 34

#### Which of the following is equivalent to 4 × 4 × 4 × 4?

An exponent is a method of indicating repeated multiplication. For example, 4 × 4 × 4 × 4 can also be expressed as 44, which is also known as “4 to the fourth power.” The small number written slightly above and to the right of a number is the exponent, and it indicates the number of times you multiply the number it accompanies by itself.

#### Which of the following is a prime number?

A prime number is a whole number that can be divided evenly by itself and by 1 but not by any other number. 2 is a prime number because its only factors are 1 and 2. 27 has factors of 1, 3, 9, and 27; 9 has factors of 1, 3, and 9; and 12 has factors of 1, 2, 3, 4, 6, and 12.

#### What is the result when 7 is added to the product of 2 and 3?

To get your desired answer of 2(3) + 7 = 13 you will need to find out which mathematical operation you should perform first. Use the acronym PEMDAS (parentheses, exponents, multiplication, division, addition, subtraction) to determine the order of operations. In this problem, perform the multiplication operation first, followed by addition.

#### Which of the following is equivalent to 2 times the total of x plus y?

2 times the total of x plus y is 2(x + y). From there, use the Distributive Property to multiply the 2 by each term inside the parenthesis. 2(x + y) = 2x + 2y

#### The absolute value of 5 plus a number equals 10. Which of the following options is the correct solution set for the above equation?

The absolute value of 5 plus a number equals 10” should be written as |5 + x| = 10. Absolute value means the distance of a number from zero, so whenever you solve for the number inside absolute value bars, you must consider both the positive and negative values. Take away the bars and set the expression equal to both 10 and -10. 5 + x = 10 gives you x = 5, and 5 + x = -10 gives you x = -15. The solution set is x = {-15, 5}.

#### What is the absolute value of the quantity two minus seven?

The absolute value of the quantity two minus seven” should be written as |2 - 7|. Absolute value is the distance of a number from zero, and since distances cannot be negative, absolute values are always positive numbers. To take the absolute value, simplify the expression between the absolute value bars and then take the positive value of it. |2 – 7| = |-5| = 5.

#### Which of the following numbers is divisible by 3?

If the sum of the digits of a number is divisible by 3, then the number itself is divisible by 3. 8 + 5 + 0 + 1 + 9 + 4 + 5 + 7 = 39 39 divided by 3 is 13 with no remainders, so the number is divisible by 3.

#### Which of the following is true?

It is true that if a number is divisible by both 2 and 3, it is also divisible by 6. For example, 108 is divisible by 2 (108 ÷ 2 = 54) and by 3 (1 + 0 + 8 = 9 ÷ 3 = 3), and therefore is also divisible by 6.

#### Convert 0.023 to a fraction.

When a number is divided by a multiple of 10, simply move the decimal one position to the left for every zero. We can take each answer to find out which one is correct. 23/100 = 0.23 (move decimal 2 positions to the left) 23/1,000 = 0.023 (move decimal 3 positions to the left) 23/10,000 = 0.0023 (move decimal 4 positions to the left) 23/100,000 = 0.00023 (move decimal 5 positions to the left) Therefore, 0.023 converted to a fraction is 23/1,000.

#### Which of the following is an integer?

An integer is any positive or negative whole number or zero. Therefore, -9 is an integer. ¾ and 0.5 are not integers because they are not whole numbers. The square root of 42 is approximately 6.48 and therefore not an integer.

#### Solve the following equation: |7 – (24 ÷ | 3-6 |)|

Solve the problem from the inside out. |7 – (24 ÷ | 3-6 |)| = |7 – (24 ÷ | -3 |)| = |7 – (24 ÷ 3)| = |7 – 8| = |-1| = 1

#### All of the following are examples of composite numbers except which one?

A composite number is a whole number that can be divided by itself, 1, and one or more other whole numbers. 6 is a composite number because it can be divided by 1, 2, 3, and 6; 9 is a composite number because it can be divided by 1, 3, and 9; and 12 is a composite number because it can be divided by 1, 2, 3, 4, 6, and 12. A prime number is a whole number that can only be divided by itself and 1. 2 is a prime number because its factors are only 1 and 2. A prime number is a whole number that can only be divided by itself and 1. 2 is a prime number because its factors are only 1 and 2. A composite number is a whole number that can be divided by itself, 1, and one or more other whole numbers. 6 is a composite number because it can be divided by 1, 2, 3, and 6; 9 is a composite number because it can be divided by 1, 3, and 9; and 12 is a composite number because it can be divided by 1, 2, 3, 4, 6, and 12.

#### Every day, Bert spends an hour commuting to and from his office, driving at an average speed of 50 mph and taking the same route each way. How far does Bert live from his office?

Use the distance formula. Remember that his total commute is 1 hour, so one-way, it will be .5 hours. d=rt d = 50(.5) d = 25

#### Laura goes to the grocery store every 5 days and Tim goes to the same grocery store every 6 days. If Laura and Tim both went to the grocery store today, when is the next time they will both go to the grocery store on the same day?

To solve this problem, you must find the least common multiple of 5 and 6, which is the smallest multiple common to both numbers. To do this, make a list of multiples for each number until at least two multiples are on both lists. 5: 5, 10, 15, 20, 25, 30… 6: 6, 12, 18, 24, 30… The lowest common multiple of both numbers is 30.

#### How many prime numbers are there between 25 and 35, inclusive?

Prime numbers are not divisible by anything other than themselves and 1. The only prime numbers in this set are 29 and 31. 25, 30, and 35 can be excluded because they are divisible by 5. 26, 28, 30, 32, and 34 are divisible by 2. 27, 30, and 33 are divisible by 3.

#### Simplify the following:

5^{1}⁄_{3} - 7^{5}⁄_{6} + 6^{1}⁄_{4 =}

In numerical form, this problem is written as: 5 1⁄3 - 7 5⁄6 + 6 1⁄4 Change the fractions to improper fractions. 5 1⁄3 - 7 5⁄6 + 6 1⁄4 (15 x 3 + 1)/3 - (7 x 6 + 5)/6 + (6 x 4 + 1) 16/3 - 47/6 + 25/4 Convert the fractions to common denominators. (16/3)(4/4) - (47/6)(2/2) + (25/4)(3/3) 64/12 - 94/12 + 75/12 Complete the calculation. -30/12 + 75/12 45/12 Simplify the fraction. 45/12 = 3 9/12 = 3 3/4 51⁄3 - 75⁄6 + 61⁄4 Change the fractions to improper fractions. 5 1⁄3 - 75⁄6 + 6 1⁄4 (15 x 3 + 1)/3 - (7 x 6 + 5)/6 + (6 x 4 + 1) 16/3 - 47/6 + 25/4 Convert the fractions to common denominators. (16/3)(4/4) - (47/6)(2/2) + (25/4)(3/3) 64/12 - 94/12 + 75/12 Complete the calculation. -30/12 + 75/12 45/12 Simplify the fraction. 45/12 = 3 9/12 = 3 3/4

#### Which answer is equivalent to seven to the fourth power?

Seven to the fourth power is 7^{1}, which is the equivalent of (7)(7)(7)(7). Multiply to get the answer: 7(7) = 49 49(7) = 343 343(7) = 2401

#### x^{3} · x^{5} =

To multiply terms with the same base (x), leave the base alone and add the exponents. 3 + 5 = 8, so x^{3} · x^{5}= x^{8}.

#### Simplify: 7x(8 – 3) ÷ 5

Simplify by using the order of operations. It’s easy to do if you remember the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction). 7x(8 – 3) ÷ 5 = 56x – 21x ÷ 5 = 35x ÷ 5 = 7x

#### The supervisor of a small company is designing a new open office layout and thinks that employees will need 4 chairs for every 2 desks. How many office chairs will be needed for an office building with 40 desks?

In order to solve this problem, set it up as a proportion with c representing the number of office chairs needed for 40 desks. 4 chairs/2 desks = c/40 desks 4/2 = c/40 2c = 40 × 4 2c = 160 c = 80 chairs

#### Will is filling up his pool for the summer, and water from the hose flows into the pool at 6 gallons per minute. The pool holds 17,280 gallons. How long, in hours, does it take to fill the pool?

The water is flowing into the pool at a rate of 6 gallons per minute. Do the math to see how long it takes to get to 17,280 gallons: 17,280 gallons ÷ 6 gallons/minute = 2,880 minutes To convert to hours, divide by 60 minutes: 2,880 minutes ÷ 60 minutes/hour = 48 hours

#### Which of the following is equal to the square root of fifty?

The square root of fifty can be written as √50. To solve this problem, try to find a factor of 50 that is a square number. In this case, 25 × 2 = 50. Write 50 as a product of its factors 25 and 2 and then take the square root of each factor: √50 = √25 × 2 = √25 √2 The square root of 25 is 5, and the square root of 2 isn’t a whole number , so the answer is 5√2.

#### Solve the following: 39 ÷ (3 + 10) – 3 x 7

Solve by using the order of operations. It’s easy to do if you remember the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction). 39 ÷ (3 + 10) – 3 x 7 = 39 ÷ 13 – 3 x 7 = 39 ÷ 13 – 21 = 3 – 21 = -18

#### Solve the following: 5!

5! is a factorial, which is multiplication indicated by an exclamation mark. To solve, write 5! as the following: 5 x 4 x 3 x 2 x 1 = 20 x 3 x 2 x 1 = 60 x 2 x 1 = 120 x 1 = 120

#### If Carol can write 3 articles for the newspaper every 2 ½ hours, how many articles can she write in 10 hours?

Convert the mixed number to an improper fraction by multiplying the denominator by the base number and adding it to the numerator. 2 ½ = (2 × 2 + 1)/2 = 5/2 Divide 10 hours by 5/2. Remember that when dividing by a fraction, you can multiply it by its reciprocal instead. 10 ÷ 5/2 = 10 × 2/5 = 20/5 = 4 Since Carol can write 3 articles in each 2 ½ hour time period, multiply 4 by 3 to find the total number of articles she can write in 10 hours. 4 × 3 = 12 articles

#### A cooler full of sodas has 12 lemon-lime drinks and 18 gingerales. If one drink is selected at random and then a second drink is selected at random, what are the chances that both sodas will be gingerales?

There is a total of 30 sodas in the cooler, and a chance of 18/30, or 3/5, that the first choice will be a gingerale. With 17 gingerales and 12 lemon-lime drinks left in the cooler, there are 17 choices in 30 that the second choice will be a gingerale. The product of the two choices is: 3/5 × 17/30 = 51/150, or 17/50 There is a chance of 17 out of 50 that both sodas will be gingerales.

#### At a track event, there are eight different top-ranked runners. How many different ways are there for the eight runners to finish the race?

This problem can be solved by using a factorial. Factorials help you find permutation – all the ways an event might turn out. Factorials are calculated by finding the product of a whole number and all the whole numbers less than that number. 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320 Therefore, there are 40,320 possible ways the eight runners could finish the race.

#### What is the square root of 320?

To solve this problem, find a factor of 320 that is a square number. In this case, 64 × 5 = 320. Write 320 as a product of its factors 64 and 5 and then take the square root of each factor: √320 = √64 × 5 = √64 √5 The square root of 64 is 8, and the square root of 5 isn’t a whole number , so the answer is 8√5.

#### Kevin and Kara are in a pancake-eating contest. Kevin can eat three pancakes per minute, while Kara can eat 3 ½ pancakes per minute. How many total pancakes can they eat in 10 minutes?

If Kevin eats 3 pancakes in one minute, you know he eats 3(10) = 30 in 10 minutes. If Kara eats 3 ½ per minute, she eats 3.5(10) = 35 in 10 minutes. Add 30 + 35 = 65 and you have your answer.

#### Rachel is driving to visit her mother, who lives 250 miles away. How long will the drive be, round-trip, if Rachel drives at an average speed of 40 mph?

The distance from Rachel’s house to her mother’s house is 250 miles, so round-trip, the distance is 500 miles. Plug this into the distance formula: d = rt 500 = 40t 12.5 = t It takes Rachel 12.5 hours to make the round-trip drive. Multiply by 60 to find the answer in minutes. 12.5(60) = 750 minutes.

#### Convert 0.075 to a percent

To convert a decimal to a percent, move the decimal point two places to the right and add a percent sign: 0.075 = 7.5%

#### What is twelve percent as a fraction?

To change a fraction into a percent, divide by 100. 12% = 12/100 = 3/25

#### Jake’s final grade in math class is determined by the average he gets on six tests in the class. So far, Jake has earned a 97, 89, 85, 90, and 99 on the first five tests. What grade must Jake earn on the sixth test in order to get a 93 average in the class?

The average is found by dividing the total of all scores and dividing it by the number of scores. Since we know the average Jake wants (93), and we have all the scores from Jake’s tests except one, we can set up an equation, with x representing Jake’s score on the 6th test. 97 + 89 + 85 + 90 + 99 + x = 93 6 460 + x = 558 x = 98

#### Carl works for a newspaper, and wants to write 15 short articles this week. He writes 3 articles on Monday. If Carl works Monday through Friday, how many articles should he write each day for the rest of the week in order to reach his goal?

To find the answer to this problem, we must determine the average number of articles Carl needs to write each day. First, subtract the three articles he has already written on Monday from the total. 15 – 3 = 12 Carl needs to write 12 articles over the remaining 4 days, so divide 12 by 4 to find the average. 12 ÷ 4 = 3 articles per day

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