Section 3: Numbers and Operations

What is the ratio of men to women if there are 15 women and 10 men?

10:15

What is 5½ equal to? *This is a mixed number! Try turning it into an improper fraction!*

11/2

The dosage for a medication is proportional to the weight of a patient. If the dosage for a patient weighing *60 kg* is *90 mg*, what is the dosage for a patient weighing *80 kg*?

120 mg Why?: Here you have a proportion with a missing value so it would be written as 60/90 = 80/x. To find x, cross multiply and get 60x = 7200. Divide by 60 on both sides to get x alone. You end up with x = 120 (mg)

1/2 + 3/7 = ?

13/14 Why?: 1/2 + 3/7 needs a common denominator (which would be 14 as they both can divide from that!) so, 1/2 (7) and then 3/7 (2) would become 7/14 + 6/14 = 13/14 (it cannot be simplified further so it is left as it is)

Write 1.45 as a percent:

145%

Simplify: 15 x (4 + 8) - 33

147

A case of pencils contains 10 boxes. Each box contains 150 pencils. How many pencils are in the case? A. 15 B. 160 C. 1500 D. 16,000

1500

What is the ratio of men to women if there are 15 men and 10 women?

15:10

Kevin is planning a party and can host 120 people. If he sends out invitations to is friends and expects 30% to decline, what is the max number of invitations he should send?

171 Why?: We know that only 120 people can attend and that he expects 30% to decline. This means that the 120 who accept is equal to 70% of the invitations he sends. We know that 70% is equal to 70/100. We also know that 120 is just a portion of his whole amount of sent invites (a number we dont know!). So we make the equation, 70/100 = 120/x. Cross multiply and get 12000 = 70x. Simplified, that becomes x = 171. Therefore, he can send out 171 total invites as he expects 30% of those invites to decline anyway

On Tuesday, a radiology clinic had 80% of patients come in for their scheduled appointments. If they saw 16 patients. How many scheduled appointments did they have?

20 Why?: You know 80% and 16 patients. This is a proportion! We know that 80% = 80/100. What we dont know is the total amount of patients seen that day. To find that out, we put 16/x = 80/100. Then cross multiply to get 1600 = 80x. Simplify that into x=20

Convert 0.46 into a fraction:

23/50 Why? 0.46 = 46/100, then divide (by 2 in this case) to get the simplest form of the answer

A smart HDTV that originally cost $1500 is on sale for 45% off! What is the *sale price* for the item? *Note: It is not asking for what the 45% is equivalent to. It is asking for what the price will be after taking that percentage OFF*

$825 Why?: Again, this is a proportion! We know 45% = 45/100. We also know the original (whole) price of 1500. What we do not know is the *sale* (part) amount that we get off. To find out, write 45/100 = x/1500. Cross multiply and get 67500/100 which simplifies into 675. This is your percentage OFF. Do not forget to take this amount away from the original (1500-675) to get a remaining sale price of $825.

²√a⁵ is the equivalent to what?

(²√a)⁵ = a⁵÷²

Place the following from LEAST to GREATEST: 2/5, -0.7, 0.35, -3/2, 0.46

-3/2, -0.7, 0.35, 2/5, 0.46 Why?: It is helpful to place them all in the same format, for this one decimals can be easily organized. In decimal form the values are as shown: 2/5 = 0.4 -0.7 = -0.7 0.35 = 0.35 -3/2 = -1.5 0.46 = 0.46 Take the resulting decimals and organize them in proper order

*For your Knowledge* In the base 10 decimal system, the numbers to the right of the decimal place end in what?

-ths

What is 15% written as a decimal?

0.15

a⁰ is the equivalent to what?

1

Which value is the smallest? 2⁰, 2∧½, 2⁻², 2¹

2⁻² Why? 2⁰ = 1 2∧½ = √2 = 1.4 2⁻² = 1/2² = 1/4 2¹ = 2

If I have a ratio of 3:4, how would it be written as a fraction?

3/4

Which of these numbers is the greatest? A. 2 3/4 B. 26/16 C. 3 1/2 D. 35/8

35/8

How is 0.36 written as a fraction and as a percent?

36/100 = 36%

Simplify: -(3)² + 4(5) +(5-6)² - 8

4

Write the expression (3.8 x 10³) + (4.7 x 10²) correctly into scientific form:

4.27 x 10³ Why?: Remember to ensure the n values (aka exponents) are matching! To do this, move the decimal in 4.7 over one time to change the exponent from 10² to 10³. Then you will have (3.8 x 10³) + (.47 + 10³). From there you simply add: 03.8 +10³ 0.47 +10³ ______________= 4.27 + 10³

What is the square root of 48? A. 16√3 B. 4√3 C. 3√4 D. 2√12

4√3 Why?: √48 = √16x3 = √16√3 = 4√3

Ari and Teagan ordered a pizza. Ari has 1/4 of his pizza left and Teagan has 1/3 of her pizza left. How much total pizza do they have?

7/12

What is 927,000,000 written in scientific notation?

9.27 x 10⁸

In a theater there are 4500 lower seats and 2000 upper seats. What is the ratio of lower seats to total seats *Tip: Be sure to pay careful attention to wording! Do not rush through the question!*

9/13 Why?: 4500:6500 is the ratio of lower to *total* seats. From there, simplify by 100 to get 45:65 and further by 5 to get 9:13. Written as a fraction this equals 9/13

Which of the following is equivalent to 0.45? A. 9/100 B. 45/100 C. 9/20 D. 4/5

9/20 Why? 0.45 = 45/100, then divide (by 5 in this case) to get the simplest form of 9/20

What is a fraction?

A fraction is a part of a whole

What is an improper fraction?

A fraction where the numerator is greater than or equal to the denominator

What is a proper fraction?

A fraction whose denominator is greater than its numerator.

What is a mixed number?

A mixed number is a fraction that includes a whole number and a fraction.

What is a proportion?

A statement of equality between two or more ratios.

When the base is the same and both have exponents, such as aⁿaⁿ, what do you do with the exponents?

Add them and THEN perform the exponential multiplication

What are the four basic arithmetic operations?

Add, subtract, multiply and divide

The following words indicate which arithmetic operation? Sum, altogether, in total, all, give, increase, increased

Addition

What is the formula for percent change?

Amount of change / original amount

What is the formula for original amount?

Amount of change / percent change

How can a percent be turned into a decimal? *Percent = Part of 100*

By dividing by 100

Adding or subtracting fractions requires a what?

Common denominator

Ratios, like fractions, can be reduced by dividing by what?

Common factors

What is a ratio?

Comparison of two quantities

What should you do when dividing in scientific notation?

Divide the *a* values and subtract the *n* values (exponents)

The following words indicate which arithmetic operation? Per, each/every, distributed, separated amongst, average, how many for each?, How many groups?, split

Division

When dividing two different bases that share a NEGATIVE exponent, such as (a/b)⁻ⁿ, what should you do?

Flop the two bases, place the exponent over each base, perform the exponential multiplication and then divide

When asked to order numbers and values from least to greatest (or vice versa) it is helpful to convert all values into the same _____________.

Format

Given 5,124.068 What decimal place is the *1* in?

Hundreds

Given 5,124.068 What decimal place is the *6* in?

Hundredths

How do you divide fractions?

Invert the second fraction (reciprocal) and multiply the tops and bottoms

What is an exponent?

It is a shorthand way of indicating that a number is multiplied by itself. The exponent tells you how many times.

If an exponent is negative, such as 5⁻³, what happens to your answer?

It will be placed as the denominator

What is scientific notation?

Method of representing very large or small numbers in the form of a x 10ⁿ, where a is a value between 1-10 and n is a nonzero integer

The following words indicate which arithmetic operation? Product, times, of, each/every, groups of, twice

Multiplication

What should you do when multiplying in scientific notation?

Multiply the *a* values and add the *n* values (exponents)

When a base with an exponent is being multiplied by another exponent such as, (aⁿ)ⁿ, what should you do with your two exponents?

Multiply the exponents by each other first and then perform the exponential multiplication

How do you multiply fractions?

Multiply the tops, then multiply the bottoms, always reduce your answer to the lowest possible point

What is the cross product method?

Multiplying the numerator to the opposing denominator

Is this proportion true? 1/3 = 5/12

No

What is a factor?

Numbers you multiply together to get another number

Given 5,124.068 What decimal place is the *4* in?

Ones

What is the formula for amount of change?

Original amount / percent change

What is PEMDAS order of operations?

Parentheses, Exponents, Multiplication, Division, Addition, Subtraction

When multiplying two different bases that share an exponent, such as (ab)ⁿ, what should you do?

Place the exponent over each base and then multiply

When dividing two different bases that share an exponent, such as (a/b)ⁿ, what should you do?

Place the exponent over each base, perform the exponential multiplication and then divide

convert decimal to fraction

Place the numbers that come after the decimal as the numerator and the corresponding base power of 10 as the denominator

When dividing two of the same bases that have exponents, such as a⁴/a⁷, what should you do with the exponents?

Subtract the two exponents and then perform the exponential multiplication

The following words indicate which arithmetic operation? Minus, less than, take away, decreased, difference, how many left?

Subtraction

What is percent change?

The amount a quantity increases or decreases from its original is the percent of change

What is the denominator?

The bottom number in a fraction

How could you determine if the following proportion is true? 2/x = 1/4 *Do not write the answer, just explain what method you might use to find the x value and determine if the proportion is true*

The cross product method

What is a Coefficient?

The number in front of the variable in a term

What is the cube root? *Hint: 3 x 3 x 3 = 27, the cube root is 3*

The number that gets multiplied by itself 2 times to make another number

What is the fourth root? *Hint: 2 x 2 x 2 x 2 = 16, the fourth root is 2*

The number that gets multiplied by itself 3 times to make another number

What is a square root? *Hint: 7 x 7 = 49, the square root is 7*

The number that gets multiplied by itself one time to make another number

What is a base?

The number that is being multiplied by itself

What is the numerator?

The top number in a fraction

Given 5,124.068 What decimal place is the *5* in?

Thousands

What should you do when adding or subtracting in scientific notation?

Write ALL numbers with the same *n* value and add or subtract the *a* values

Is this proportion true? 1/2 = 5/10

Yes

Is this proportion true? 2/10 = x/5

Yes Why?: Using cross multiplication, you would have 10 = 10x. To find X, you divide by 10 in this case and end up with x=1. Replace the X in the original equation with your newfound number and see if the proportion matches. 2/10 = 1/5 is true!

What is ⁿ√a/b equivalent to?

ⁿ√a ______ ⁿ√b

What is ⁿ√ab equivalent to?

ⁿ√aⁿ√b