## Fractions and Decimals, Mixed Numbers, Improper Fractions, Ratios

**Fractions**

*A fraction represents a whole number that has been divided into equal parts.*

Example 1: If you have a pie and you cut it into 4 equal slices, 1 of those slices is written as 1/4

*A fraction can also represent a part of a group. *

Example 2: If you have 5 pieces of candy and give 2 to your friend, those 2 candies are written as 2/5

A fraction is made up of two parts, the **numerator **and the **denominator**.

The **numerator** is the top number of the fraction. It is the number that represents the part of the whole.

The **denominator** is the bottom number of the fraction. It is the number that represents the total amount.

Fraction Addition and Subtraction

Step 1: Make sure the bottom numbers (denominators) are the same

if they are not, find the least common multiple (LCM) of all the denominators and rewrite the fractions so they all have the LCM as their denominators

Step 2: Add the numerators (for addition) and subtract the numerators (for subtraction)

Step 3: Simplify the fraction (if necessary)

Fraction Multiplication and Division

*Multiplication*:

Multiply the numerators of the fractions to get the new numerator

Multiply the denominators of the fractions to get the new denominator

Simplify the terms (if necessary)

*Division*:

Flip the second fraction’s numerator and denominator (find the reciprocal) and perform fraction multiplication

Decimals:

A decimal number is a number that uses a decimal point to express a value that has a whole number part and a fractional part, separated by a decimal point. It is based on the number 10.

The first place value after the decimal is the tenths place (it is not the “one”ths place!)

As you move right (from the decimal point), each place value position is 10 times smaller

As you move left (from the decimal point), each place value position is 10 times larger

Decimals can be rewritten as fractions.

2.3 = 23/10

13.76 = 1376/100

0.8 = 8/10 = ⅘

0.43 = 43/100

2.3 = 2 and 3/10

13.76 = 13 and 76/100

Decimal Addition and Subtraction

Write down the numbers, one under the other, with the decimal points lined up

Put in zeros so that the numbers have the same length

Add using column addition (remember to include the decimal point in your answer)

*Subtraction uses the same method (just subtract!)

Decimal Multiplication

Multiply normally, ignoring the decimal points

Count the number of decimal places in the two decimals and add them together

Put the decimal point in the answer based on the number of places from Step 2

**Mixed Numbers**

A mixed number is a number consisting of a whole number (integer) and a fraction.

**Comparing Mixed Fractions**

Compare the whole number. If one value has a smaller whole number than the other, then that value is the smaller number.

For example, 1 1/2 is smaller than 2 1/4 because 1 is less than 2.

If the whole numbers are the same, compare the fractions ( 1 1/4 < 1 1/2 because 1/4 is less than 1/2)

**Improper Fractions**

An improper fraction is a number where the numerator is greater than or equal to the denominator, like 5/2 or 7/3 (a fraction that is greater than one).

On the other hand, a proper fraction is a number where the numerator is less than the denominator (a fraction that is less than one).

Converting Mixed Numbers to Improper Fractions

Multiply the whole number by the fraction’s denominator

Add the result to the numerator

Write the result on top of the original denominator

Converting Mixed Numbers to Improper Fractions

Multiply the whole number by the fraction’s denominator

Add the result to the numerator

Write the result on top of the original denominator

**Ratios**

A ratio compares two things, using “:”, the word “to” or as a fraction. You can write ratios as fractions and decimals. Make sure to **simplify the ratios.**

Word Problem with Ratios

A class of 32 students has 12 girls. What is the ratio of girls to boys?

The problem gives the number of girls and the total number of students

To find the number of boys subtract the number of girls from the total

32 students - 12 girls = 20 boys

The question asks for girls to boys

12 girls : 20 boys (make sure it is in the same order, girls to boys)

Simplify

**Answer:**The ratio of girls to boys is 3 girls : 5 boys

"Part-to-Part" and "Part-to-Whole" Ratios

Ratios can represent the relationship between two parts, or a part to the whole.

Example: There are 5 marbles, 2 are green, and 3 are blue

*Part-to-Part:*

The ratio of green to blue marbles is 2:3 or 2/3

The ratio of blue to green marbles is 3:2 or 3/2

*Part-to-Whole:*

The ratio of green marbles to total marbles is 2:5 or 2/5

The ratio of blue marbles to total marbles is 3:5 or 3/5

### Take this practice quiz to see if you have mastered the content!

### After you have taken the quiz, click the bar to reveal the answers.

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**If you want more practice, check out these links!**

Decimal: https://www.varsitytutors.com/hotmath/hotmath_help/topics/operations-with-decimals

Fraction operations practice: https://www.hec.ca/en/cams/help/topics/Fractions_operations.pdf

Converting between improper fractions and mixed fractions: https://www.youtube.com/watch?v=6dd-cjZenNE

https://www.youtube.com/watch?v=Z3sH6i-l1vE (mixed to improper)

https://www.youtube.com/watch?v=11g8LDC5tWU (improper to mixed)

mathisfun.com has all the topics (very good resource for simple explanations)