Fractions, Decimals, Percents

Question 1

Answer 1

For adding & subtracting fractions, first find a common denominator (Least Common Multiple of 3 and 5)

Question 2

Answer 2

Question 3

Answer 3

Question 4

Answer 4

Question 5

Answer 5

Multiply horizontally:

Question 6

Answer 6

Dividing fractions: Flip the 2nd fraction and multiply

Question 7

Answer 7

Question 8

Answer 8

When we take the RECIPROCAL, we flip the fraction

This is the same as 1 divided by the fraction.

Question 9

Write “z is x percent of y” as an equation

Answer 9

Percent means “divide by 100”

“of” means “multiply”

“is” means =

Question 10

Write “x is 20% greater than y” as an equation

Answer 10

Question 11

Write “x is 15% less than y” as an equation,

a) write using a decimal
b) convert to a fraction

Answer 11

Question 12

Write “x is n percent greater than y” as an equation

Answer 12

Question 13

A bike was $50, and is now on sale for $32. What is the percent change in price?

Answer 13

Question 14

Distribute:

-3(x-7)

Answer 14

= -3*x + -3*-7

= -3x + 21

Be careful with your SIGNS! Double negative turns into a positive

Question 15

Factor out the Common Factor:

4x + 4y =7

Answer 15

Both terms have “4” as a factor.

So, we can factor it out (this is the inverse process to “distribution”)

4(x+y) = 7

Question 16

Factor out the Common Factor:

3x + 6y + 15z = 7

Answer 16

Factor out the Common Factor:

The common factor between 3, 6, and 15 is 3.

(we can prove this by writing the prime factorization: 6=2*3, 15=5*3)

We factor out 3 from each “term”.

3(x + 2y + 5z)

Question 17

Multiplying Exponents:

What is 2³ * 2⁴ ?

Answer 17

2⁷

When we multiply exponents with the same bases, we add the powers

a^b * a^c = a^b+c

Question 18

What is 2 * x³ * y² * 3 * y⁴ * x ?

Answer 18

We match up what can be simplified:

2*3 = 6

x³ * x = x⁴

y² * y⁴ = y⁶

6x⁴y⁶

Question 19

Distribute: -2x(-3x + 5)

Answer 19

• 2x * -3x = 6x²
• 2x * 5 = -10x

6x² - 10x

Question 20

Factor out the Common Factor:

15x² - 10x

Answer 20

This time, we can factor out both a number and a variable:

The common prime factor of 15 and 10 is 5.

The common variable is x

So, we factor out 5x:

5x(3x-2)

Question 21

Scientific Notation:

Write out as a number: 2.3 * 10⁷ = ?

Answer 21

23,000,000

* 10⁷ is like multiplying by 10, 7 times. This moves the decimal point 7 places to the right.

Question 22

What is the units digit, tens digit, and hundreds digit of the number 5318?

Answer 22

units: 8
tens: 1
hundreds: 3

Question 23

What is the units digit of 32¹⁹ ?

Answer 23

We need to find the pattern.

32¹⁹ will have the same units digit as 2¹⁹

2¹ = 2

2² = 4

2³ = 8

2⁴ = 16

2⁵ = 32

2⁶ = 64

The pattern repeats for every 4 numbers in the pattern: {2, 4, 8, 6}

Every 4th power will have a units digit of 6: 2⁴, 2⁸, 2¹², 2¹⁶

So, 2¹⁷ ends in 2, 2¹⁸ ends in 4, 2¹⁹ ends in 8.

Question 24

What is the units digit of 7¹⁴ ?

Answer 24

7¹ =7

7² = 49

Only the units digit will affect the next power: so we can do 9*7 =63 –> 7³ ends with 3

3*7 =21 –> 7⁴ ends with 1

1*7 = 7 –>7⁵ ends with 7.

So, the pattern repeats for every 4 numbers in the pattern: {7, 9, 3, 1}

So, the 4th, 8th, and 12th powers would have units digit of 1.

7¹³ has units digit of 7

7¹⁴ has units digit of 9

Question 25

If x and y are integers, with y > 1, how do we determine whether the fraction x/y is a repeating decimal (for example, 1/3 = .33333333… repeating), versus a “Terminating Decimal” with a finite number of digits (example: 1/40 = 0.025)?

Answer 25

The numerator doesn’t matter.

If the denominator only includes the prime factors 2 and/or 5, it is not repeating.

Example: 1/200 = ?

200 = 2*2*2*5*5 —> so it will be a “Terminating Decimal”

1/200 = 0.005

1/60= 3*2*2*5 = .01666666…… —> “Repeating Decimal”

Question 26

For every 5 men, there are 3 women.

Express the above ratio in 2 ways

Answer 26

The ratio of men to women is:

5 : 3 or

5/3

Note: The order of wording matters! the ratio of women to men is 3:5

Be mindful to prevent this “Backwards” error.

Question 27

If there are 25% more women than men, what is the ratio of men to women?

Answer 27

Question 28

If the ratio of men to women is 5:7, what fraction of the total are women?

Answer 28

Question 32

If the ratio of men to women is 3:4, which of the following could NOT be the total number of men and women combined?

A) 35

B) 72

C) 49

D) 21

E) 280

Answer 32

We can’t have a fraction of a person.

Therefore, the total must be a multiple of 7. (3+4)

7, 14, 21, 28, etc.

Only B is not a multiple of 7. (we might notice that 70 is a multiple of 7, so 72 isn’t)

35= 5*7

49=7*7

21= 3*7

280= 40*7

Question 33

If the ratio of flour to water in a recipe is 3:2, how many cups of flour would we need for 7 cups of water?

Answer 33

Unlike for people, We CAN have fractions of units.

We can use the Unknown Multiplier method:

The ratio of flour to water is 3x : 2x, with x as our unknown multiplier.

We know the amount of water, 7, so 2x=7.

Solve for x: x=3.5

Now plug in x=3.5 and find the flour:

3x = 3*3.5 = 10.5 cups of flour

Question 34

If the ratio of dogs to cats to birds is 2:3:5, and there are 24 cats, how many total animals are there?

Answer 34

We can use the unknown multiplier:

2x:3x:5x

3x represents cats. 3x=24

x=8

So, our multiplier is 8.

Counting all 3 animals, we have 2x+3x+5x = 10x

So, 10x = 10*8 = 80

Question 35

There are 3x for every 2y.

There are 3z for every 5x

What is the ratio of y to z?

Answer 35

ratio 1) 3x:2y

ratio 2) 3z:5x

We use a similar process to finding a common denominator with fractions:

What is the common factor between 3x and 5x? 15x

Multiply the 1st ratio by 5: 15x:10y

Multiply the 2nd ratio by 3: 9z:15x

So, for every 15x there is 10y, and 9z

Therefore the ratio of y to z is 10:9

Question 36

What is the definition of “directly proportional”?

Answer 36

When 2 quantites vary by a constant ratio.

It is a linear relationship: y = mx, where m is the slope or constant ratio The graph would start at 0,0

examples: someone earns $20 per hour, with no other payments.