ACT Math

Question 1

How do you count how many consecutive integers are from 15 through 52?

Answer 1

To count how many consecutive integers from 15-52, subtract the smallest consecutive integer from the largest and add 1. E.g. For 15 through 52, it would be 52-15=37 plus 1. The answer is 38.

Question 2

What are numbers that are relatively prime? e.g. Are 35 and 54 relatively prime?

Answer 2

Two relatively prime numbers are integers that have no common factor other than 1. To determine this, break down both numbers to their prime factorizations. If they have no prime factors in come, they are relatively prime. E.g. 35=5x7 and 54=2x3x3x3. They have no prime factors in common, so 35 and 54 are relatively prime.

Question 3

How do you know if a number will be a multiple of 2 or 4? e.g. 558?

Answer 3

Numbers are divisible by 2 if the last digit is an even number. Numbers are divisible by 4 if the last two digits form a multiple of 4. E.g. the last two digits of 558 is 58, which is not divisible by 4, so 558 is not a multiple of 4.

Question 4

How do you add and subtract fractions? e.g. 3/4 + 1/5= ?

Answer 4

To add and subtract fractions, find a common denominator for each fraction. Then add or subtract the numerator. E.g. 3/4 + 1/5 = 15/20 + 4/20 = 19/20.

Question 5

How do you multiply fractions? e.g. 3/8 x 1/2 = ?

Answer 5

To multiply fractions, simply multiply the numerators and multiply the denominators. E.g., 3/8 x 1/2= 3/16

Question 6

How do you divide fractions? e.g. 1/2 divided by 3/4 = ?

Answer 6

To divide fractions, change the division signal to multiplication, invert the second fraction, and multiply. E.g. 1/2 divided by 3/4= 1/2 x 4/3=4/6 or reduced to 2/3

Question 7

How do you convert an improper fraction to a mixed number? e.g. convert 212/9 =

Answer 7

To convert an improper fraction to a mixed number, divide the denominator into the numerator to get a whole number quotient with a remainder. The quotient becomes the whole number part of the mixed number and the remainder is the new numerator with the same denominator. (You can find the remainder by multiplying the whole number by the denominator and the noting the difference wit the original numerator. e.g. 212/9= 23.555. If you multiple 23x9=207, and 207-212=5….so the remainder is 5. The answer is 23 5/9.

Question 8

How do you compare the values of two or more fractions? e.g. 4/9 and 5/8?

Answer 8

To compare the values of two or more fractions, change the fractions so that you have a common denominator OR convert both fractions to decimals. E.g. 4/9=.444 and 5/8=.625

Question 9

How do you tell if a number is a multiple of 3 or 9? e.g. 378

Answer 9

A number is divisible by 3 if the SUM of its digits is divisible by 3. A number is divisible by 9 if the sum of its digits is divisible by 9. E.g. The sum of the digits 378 is 18, which is divisible by both 3 and 9. Therefore, 378 is divisible by 3 and by 9.

Question 10

What is prime factorization?

Answer 10

Reduce the number by factors until only prime numbers remain. E.g. Prime factorization of 56= 7*8=7*2*4=7*2*2*2=7*2^3

Question 11

How do you simplify a fraction to the lowest terms? E.g. 36/56=?

Answer 11

To reduce a fraction to lowest terms, factor the numbers and then cancel all factors the numerator and denominator have in common. E.g. 36/56=3*4*3/4*2*7=3*3/2*7=9/14

Question 12

How do you add mixed numbers? E.g. 1 2/3 + 1 1/5=?

Answer 12

To add mixed numbers, convert the fraction of the mixed numbers to an improper fractions with the same denominators. Add the whole numbers and add the fractions. You can then reduce the improper fraction back to a mixed number if needed. E.g. 1 2/3 + 1 1/5 = 1 10/15 + 1 3/15= 2 13/15.

Question 13

How do you convert a fraction to a decimal? e.g. 3/8=?

Answer 13

To convert a fraction into a decimal, divide the numerator by the denominator. E.g. 3/8=.375

Question 14

How do you convert a decimal to a fraction? e.g. .625

Answer 14

To convert a decimal to a fraction, put the decimal over 1 and multiply the numerator and the denominator by 10 raised to the number of digits to the right of the decimal. E.g. To convert .625 to a fraction, multiply .625/1 by 10^3/10^3 or 1,000/1,000. Then simplify. 625/1000=5/8.

Question 15

How does this common decimal convert into fraction form? .1= ?

Answer 15

This common decimal converts into fraction form: .1= 1/10

Question 16

How does this common decimals convert into fraction form? .125=?

Answer 16

This common decimal converts into fraction form: .125=1/8

Question 17

How does this common decimal convert into fraction form? .2=?

Answer 17

This common decimal converts into fraction form: .2=1/5

Question 18

How does this common decimal convert into fraction form? .25=?

Answer 18

This common decimal converts into fraction form: .25=1/4

Question 19

How does this common decimal convert into fraction form? .3333=?

Answer 19

This common decimal converts into fraction form: .3333=1/3

Question 20

How does this common decimal convert into fraction form? .375=?

Answer 20

This common decimal converts into fraction form: .375=3/8

Question 21

How does this common decimal convert into fraction form? .5=?

Answer 21

This common decimal converts into fraction form: .5=1/2

Question 22

How does this common decimal convert into fraction form? .625=?

Answer 22

This common decimal converts into fraction form: .625=5/8

Question 23

How does this common decimal convert into fraction form? .6667=?

Answer 23

This common decimal converts into fraction form: .6667=2/3

Question 24

How does this common decimal convert into fraction form? .75=?

Answer 24

This common decimal converts into fraction form: .75=3/4

Question 25

How does this common decimal convert into fraction form? .875=?

Answer 25

This common decimal converts into fraction form: .875=7/8

Question 26

What formula should you use to find a percent? Examples: What is 8 percent of 25? 20 is 6 percent of what number? 36 is what percent of 54?

Answer 26

To find the percent of a number, multiply: part=percent x whole Examples: What is 8 percent of 25? part=.08 x 25 = 2 20 is 6 percent of what number? 20= .06x whole, whole=333.33 36 is what percent of 54? 36= percent x 54, percent=67%

Question 27

How do you increase or decrease a number by a certain percentage? Examples: Increase 50 by 30%? Decrease 25 by 15%

Answer 27

To increase or decrease a number by a certain percentage, add/subtract the percent to/from 100 and convert to a decimal, then multiply by the number. Examples: Increase 50 by 30%? 100%+30%=130% or 1.3. Multiply 50x1.3=65 Decrease 25 by 15% 100%-15%=85% or .85. Multiply 25x.85=21.25

Question 28

How do you find the original value before it was increased or decreased by a certain percentage? Example: After a 15 percent increase, the sales were $1,250. What were the sales before the increase? After a 5 percent decrease, the population dropped to 3,325. What was the population before the decrease?

Answer 28

To find the original value before it was increased or decreased by a certain percentage, plug in the numbers you know: original value x (1.0 +/- percentage change)=after value Example: After a 15 percent increase, the sales were $1,380. What were the sales before the increase? original x (1 + .15)= $1,380 original sales = $1,200 After a 5 percent decrease, the population dropped to 3,325. What was the population before the decrease? original x (1 - .05)= 3,325 original population = 3,500

Question 29

When you want to find out the total change after being given a series of percent increases and/or decrease, how do you determine the total change? E.g. The price increased 10% in year 1, and 15 percent in year 2. What was the combined % increase?

Answer 29

To find out the total change after being given a series of percent increases and/or decrease, DO NOT simply add the percents. You must set up multiple equations and start with the ending of the previous increase/decrease. E.g. The price increased 10% in year 1, and 15% in year 2. What was the combined % increase? Year 1: 1.00 + .10= 1.10 Year 2: 1.10 + (.15 of 1.10)= 1.10+.165=1.265 or 26.5% increase after year 2

Question 30

What is the ratio of 25 blue to 15 red?

Answer 30

To find the ratio of 25 blue to 15 red, set up the fraction 25/15 or 5/3.

Question 31

How do you convert part-to-part ratios to part-to-whole ratios? E.g. If the ratio of girls to boys in class is 3 to 2, what is the ratio of girls to the whole class? boys to whole class?

Answer 31

To convert part-to-part ratios to part-to-whole ratios, put each number over the sum of the numbers. E.g. If the ratio of girls to boys in class is 3 to 2, what is the ratio of girls to the whole class? girls to class: 3/ (3+2)= 3/5 boys to class 2/ (3+2)= 2/5

Question 32

How do you find the proportion? E.g. x/6=3/4

Answer 32

To solve a proportion, cross multiply. E.g. x/6=3/4 4x=3x6=18 x=18/4=4.5

Question 33

How do you solve a problem involving rates? E.g. If snow falls at the rate of 1 foot every 4 hours, how many inches will it fall in 7 hours?

Answer 33

To solve a problem involving rates, make sure you convert units, if needed, and cross multiply. E.g. If snow falls at the rate of 1 foot every 4 hours, how many inches will it fall in 7 hours? 1 foot/4 hours= 12 inches/4 hours x inches/7 hours 4x=12 * 7 x=21 inches

Question 34

How do you quickly find the average of a series of evenly spaced numbers? E.g. Find the average of all numbers 21 through 79.

Answer 34

To quickly find the average of a series of evenly spaced numbers, just average the smallest and largest number. E.g. Find the average of all numbers 11 through 79. (11+79)/2=90/2=45

Question 35

How do you find the sum of all of the numbers in a series? E.g. In the series 13 through 55, what is the sum of these numbers?

Answer 35

To find the sum of all of the numbers in a series, find the average and then multiply by the number of terms in the series. E.g. In the series 13 through 55, what is the sum of these numbers? (13+55)/2=68/2=34 number of terms=(55-13)+1=43 so average 34 x 43 terms =1,462 is the sum

Question 36

How do you find the missing number in a series when you are given the average? E.g. The average of 4 numbers is 8. The numbers are 3,7,9, and ?

Answer 36

To find the missing number in a series when you are given the average, calculate the sum by multiplying the average by the number of terms, and then back into the missing number. E.g. The average of 4 numbers is 8. The numbers are 3,7,9, and ? 4x8=32 The terms 3+7+9= 19. So 32-19=13. The missing number is 13.

Question 37

How do you simplify a square root? E.g. square root of 12= ?

Answer 37

To simplify a square root, factor out perfect squares under the radical, and take them outside the radical. E.g. square root of 12= ? square root of 3x 4= 2 square root of 3

Question 38

How do you add and subtract roots?

E.g. 2 square root of 3 + 3 square root of 3= ?

Answer 38

To add and subtract roots, they must have the same radical/root.

E.g. 2 square root of 3 + 3 square root of 3= 5 square root of 3

Question 39

How do you factor the difference of two squares?
E.g. a²-b²= ?

Answer 39

To factor the difference of two squares, use the formula:

a²-b²= (a+b)(a-b)

E.g. x²-16 = (x+4)(x-4)

Question 40

How do you simplify an algebraic fraction?
E.g. x²-x-12 / x²-9 = ?

Answer 40

To simplify an algebraic fraction, find common factors and cancel them out.
E.g. x²-x-12 / x²-9 = (x-4)(x+3) / (x-3)(x+3)= (x-4) / (x-3)

Question 41

How do you solve a quadradic equation?
E.g. x²+12=7x solve for x?

Answer 41

To solve a quadradic equation, rewrite the equation to = zero, factor and determine which solutions = zero.
E.g. x²+12=7x solve for x?

Rewrite: x²- 7x + 12 = 0
Factor: (x-3) (x-4) = 0
Solve: x= 3 and x=4 will give zero.

Question 42

What is the Triangle Inequality Theorem?

Answer 42

Triangle Inequality Theorem states that the length of one side of a triangle must be greater than the difference and less than the sum of the lengs of the other two sides of the triangle.

E.g. If one side is 3 and another side 7, then the length of the 3rd side must be either greater than 7-3=4 and less than 7+3=10.

Question 43

What are isosceles triangles?

Answer 43

Isosceles triangles are triangles with 2 equal sides and 2 equal angles.

Question 44

How do you find the area of a sector?

E.g. A circle has a radius of 4, the sector’s angle is 90 degrees.

Answer 44

To find the area of a sector, find the proportion and multiply by π r ²

E.g. A circle has a radius of 4, the sector’s angle is 90 degrees.

90/360= 1/4 * π *4^{2 =} 4π area

Question 45

How do you find the surface area of a rectangular solid?

E.g. How much wrapping paper do you need to cover a box that is 3x5x2?

Answer 45

To find the surface area of a rectangular solid, use the formula:

Surface area = 2 lw x 2 wh x 2 lh

E.g. How much wrapping paper do you need to cover a box that is 3x5x2 inches?

2 (3*5)+2 (3*2) + 2 (5*2) = 30+12+20= 62 square inches

Question 46

When 3/13 is written as a decimal, what is the 100th digit after the decimal point?

Answer 46

When 3/13 is written as a decimal, to find the 100th digit after the decimal point, divide until you see the pattern of where the 100th digit is.

3/13= .23076923 - there are 6 digits to the right of the decimal before it repeats. So 100/6 = 96 4/6 …so the 100th digit is in the “4th position to the right” i.e. “7”

Question 47

Jim’s average score after 3 tests is 17. What score would Jim need on the next test to bring his average to 19?

Answer 47

Jim’s average score after 3 tests is 17. To find the score would Jim need on the next test to bring his average to 19, use the old average to find the total sum of the first 3 tests. Use the new average to find the total sum of the 4 tests. The difference between those sums is the score he needs

3x17=51 sum after 3 tests

4x19=76 sum after 4 tests

76-51=25 needed

Question 48

In 2000, there were 160 1st graders in school A and 200 1st graders in school B. School A is adding 4 new 1st graders each year, and school B’s 1st graders are shrinking by 6 a year. In what year will school A and school B have the same number of 1st graders?

Answer 48

In 2000, there were 160 1st graders in school A and 200 1st graders in school B. School A is adding 4 new 1st graders each year, and school B’s 1st graders are shrinking by 6 a year. In what year will school A and school B have the same number of 1st graders?

To solve:

The initial difference is 200-160= 40 kids
The gap changes by 4- (-6) =10 kids each year
It will take 40/10 or 4 more years until they are equal – that would be the year 2004.

Question 49

How do you find the slope of the line with the equation:

2x-4y=5

Answer 49

To find the slope of the line with the equation:

2x-4y=5

Put the equation into the y= mx + b format.

-4y= -2x + 5

y= 1/2 x - 5/4 so the slope is 1/2.

Question 50

Judy reads at a rate of 150 words per minute. How long will it take Judy to read 1000 words?

Answer 50

Judy reads at a rate of 150 words per minute. How long will it take Judy to read 1000 words?

Use the proportion -

150 words/minute =1000 words/ x minute

cross multiply 150x = 1000 x= 6.6667 or 6 mins 40 seconds

Question 51

If a triangle has sides 6, 8 and 10, and a square has a perimeter of 28, what is the difference in area between the triangle and the square?

Answer 51

If a triangle has sides 6, 8 and 10, and a square has a perimeter of 28, what is the difference in area between the triangle and the square?

The triangle 6-8-10 is a right triangle (3-4-5) and has an area of 1/2*6*8=24. The square has sides of 28 / 4 = 7 and an area of 7*7= 49. The difference between the area of the triangle 24 and area of the square 49 is 25.

Question 52

If the equation x²+kx+36=0 has exactly only one real solution for x, which of the following could be the value of k?

a. -6
b. 0
c. 6
d. 12
e. 18

Answer 52

If the equation x²+kx+36=0 has exactly only one real solution for x, which of the following could be the value of k?

Factor the equation: x²+ kx + 36 = 0 (x + or - 6)²= x² + or - 12x + 36. The coefficient of the middle term is either +12 or -12. Only 12 or “D” fits.

Question 53

For all x, (3x+4)(4x-3)= ?

Answer 53

For all x, (3x+4)(4x-3)= 12x²-9x+16x-12 = 12x²+7x-12

Question 54

What is the solution for in the system of equations below? What is x?
3x+4y=31

3x-4y= -1

a. 4
b. 5
c. 6
d. 9
e. 10

Answer 54

What is the solution for in the system of equations below? In this case, you can add the 2nd equation to the 1st equation:
3x+4y=31

3x-4y= -1

________

6x =30

6x/6=30/6

x=5 The correct answer is B.

Question 55

(a+b)/c = ?

Answer 55

(a+b)/c = a/c +b/c

Question 56

How do you convert a mixed number to an improper fraction ?
E.g. A b/c = ? or 3 1/5 = ?

Answer 56

To convert a mixed number to an improper fraction,
A b/c = ((A * c) + b)/c

3 1/5 = ((3*5)+1)/5= 16/5

Question 57

What are the prime numbers?
Is 0 prime? Is 1 prime? Is 2 prime?

Answer 57

2 is the only even prime umber. O and 1 are not prime.

2, 3, 5, 7
11, 13, 17, 19
23, 29
31, 37
41, 43, 47
53, 59
61, 67
71, 73, 79
83, 89
97

Question 58

True or False?

a/ (b+c) =? a/b + a/c

Answer 58

FALSE

a/(b+c) DOES NOT EQUAL a/b + a/c

Question 59

How do you find the Greatest Common Factor of 2 numbers? e.g. 24 and 108

Answer 59

24= 12*2=4*3*2=2*2*3*2= 2³*3

108=54*2=27*2*2=9*3*2=3*3*3*2*2=2²*3³

Now circle and multiply the common factors.

Both have 2²*3=4*3=12 is the greatest common factor

Question 60

How do you find the Least Common Multiple of 2 or more numbers? e.g. 30 and 45

Answer 60

To find the Least Common Multiple, use prime factorization:

30=6*5=2*3*5
45=15*3=5*3*3

Then, multiply each factor the greatest number of times it occurs in either number.

2 - one time
3- two times
5-one time
2*3*3*3*5=90 is the least common multiple of 30 and 45.

Question 61

a (b+c) = ?

Answer 61

a (b+c) = ab+ac

Question 62

-a(b+c) = ?

Answer 62

-a (b+c) = -ab -ac

Question 63

-a (b - c) = ?

Answer 63

-a (b - c) = -ab + ac

Question 64

What does X⁰= ?

Answer 64

X⁰ = 1

Anything to the power of 0 is 1.

Question 65

a^b / a^c = ^?

Answer 65

a^b / a^c = a ^b-c

Question 66

a^c * b^c = ?

Answer 66

ac * bc = (a*b)c

Question 67

a ^-b= ?

Answer 67

a ^-b= 1 / a^b

E.g., 2^-3 = 1 / 2³ = 1 / 2*2*2 = 1 /8