# Adding and Subtracting Mixed Numbers and Improper Fractions

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Adding and Subtracting Mixed Numbers and Improper Fractions

Just like our counting numbers (1, 2, 3,β¦), fractions can also be added and subtracted. When counting improper fractions and mixed numbers, we are counting the number wholes and parts.

Note: The rules for adding and subtracting improper fractions are the same as working with proper fractions.

Case 1: Adding and Subtracting Improper Fractions with Common Denominators Step 1: Keep the denominator the same. Step 2: Add or subtract the numerators. Step 3: If the answer is an improper form, reduce the fraction into a mixed number.

Exercise 1: Add the fractions, + . Letβs draw a picture to see what this looks like.

The 4 in the denominator tells us that each whole is cut into 4 equal portions. By adding the fractions we are grouping the total number of wholes and parts.

We have 5 slices and each whole is made up of 4 slices, .

We have 6 slices and each whole is made up of 4 slices, .

Altogether, we have 2 wholes and 3 quarters,

How does the math work?

Step 1: Since the two fractions have equal sized slices, keep the denominator the same, .

Step 2: Add the numerators,

.

Step 3: Thus, we have wholes. Tutoring and Learning Centre, George Brown College 2014

www.georgebrown.ca/tlc

Adding and Subtracting Mixed Numbers and Improper Fractions

Case 2: Adding and Subtracting Improper Fractions with Different Denominators Step 1: Find the Lowest Common Multiple (LCM) between the denominators. Step 2: Multiply the numerator and denominator of each fraction by a number so that they have the LCM as their new denominator. Step 3: Add or subtract the numerators and keep the denominator the same. Step 4: If the answer is an improper form, reduce the fraction into a mixed number.

Exercise 2: Subtract the fractions,

.

Step 1: List the multiples of 6 and 8.

Multiplies of 6: 6, 12, 18, 24, 30, 36, 48β¦ Multiplies of 8: 8, 16, 24, 32, 40, 48, 56β¦

The Lowest Common Multiple between 6 and 8 is 24.

Step 2: a) We need to find a number that when multiplied to the top and bottom of , we get the LCM (24) as the new denominator.

Since

, we need to multiply the numerator and the denominator by 4.

Thus, is equivalent to

b) We need to find a number that when multiplied to the top and bottom of , we get the LCM (24) as the new denominator.

Since

, we need to multiply the numerator and the denominator by 3.

Thus, is equivalent to

Step 3: Since our fractions now have equal sized slices, we can subtract their

numerators. Thus, we now have,

of a whole.

Tutoring and Learning Centre, George Brown College 2014

www.georgebrown.ca/tlc

Adding and Subtracting Mixed Numbers and Improper Fractions

Case 3: Adding and Subtracting Mixed Numbers Method 1 Step 1: Convert all mixed numbers into improper fractions. Step 2: Check! Do they have a common denominator? If not, find a common denominator. Step 3: When necessary, create equivalent fractions. Step 4: Add or subtract the numerators and keep the denominator the same. Step 5: If the answer is an improper form, reduce the fraction into a mixed number.

Exercise 3: Subtract the fractions,

.

Step1: Convert both mixed numbers into improper fractions.

π

ππ

ππ π

π

π

ππ π

Step 2: List the multiples of 4 and 7.

Multiplies of 4: 4, 8, 12, 16, 20, 24, 28β¦ Multiplies of 7: 7, 14, 21, 28, 35β¦

The Lowest Common Multiple between 4 and 7 is 28.

Step 3: a) We need to find a number that when multiplied to the top and bottom of , we get the LCM (28) as the new denominator.

Since

, we need to multiply the numerator and the denominator by 7.

Thus, is equivalent to

b) We need to find a number that when multiplied to the top and bottom of , we get the LCM (28) as the new denominator.

Tutoring and Learning Centre, George Brown College 2014

www.georgebrown.ca/tlc

Adding and Subtracting Mixed Numbers and Improper Fractions

Since

, we need to multiply the numerator and the denominator by 4.

Thus, is equivalent to

Step 4: Since our fractions now have equal sized slices, we can subtract their

numerators. Subtracting their numerators we have,

of a whole.

Step 5: Thus, we have wholes.

Case 4: Adding and Subtracting Mixed Numbers Method 2 In this second method, we will break the mixed number into wholes and parts.

Step 1: Add or subtract the whole number part. Step 2: Check! Does the fraction part share a common denominator? If not, find one. Step 3: When necessary, create equivalent fractions. Step 4: Add or subtract the numerators of the fraction part and keep the denominator the same. Step 5: If the answer is an improper form, reduce the fraction into a mixed number.

Exercise 4: Jessica is years old today. How old was she years ago?

Since we are looking at the difference between her current and past ages, our equation

will look like,

.

Step 1: Subtract the whole number part,

Step 2: List the multiples of 2 and 4. The Lowest Common Multiple between 2 and 4 is 4.

Multiplies of 2: 2, 4, 6, 8β¦ Multiplies of 4: 4, 8, 12β¦

Step 3: a) We need to find a number that when multiplied to the top and bottom of , we get the LCM (4) as the new denominator.

Since

, we need to multiply the numerator and the denominator by 2.

Tutoring and Learning Centre, George Brown College 2014

www.georgebrown.ca/tlc

Adding and Subtracting Mixed Numbers and Improper Fractions

Thus, is equivalent to

b) Since already has the LCM (4) as the denominator, we leave the fraction as it is.

Step 4: Since our fraction part now has equal sized slices, we can subtract their

numerators. Subtracting their numerators we have,

of a whole.

Step 5: Combining our whole number and fraction parts we get,

.

Exercise 5: Subtract the fractions,

Step 1: Subtracting the whole number part, we get Step 2: Subtracting the fraction part, we get

of a whole.

Since we cannot take 3 away from 1, we need to borrow a whole from the first fraction.

Given

, letβs borrow a whole by following the steps below: Rewrite 3 wholes into 2 wholes + 1 whole.

Since each whole has 4 slices, add the four slices from the borrowed whole into the numerator of the fraction part.

Thus, we have created an equivalent fraction where =

Step 3: Now we are able to subtract the fractions,

.

Subtracting the whole number part, we are left with,

Subtracting the fraction part, we are left with,

= of a whole.

Combining our whole number and fraction parts we are left with,

. Tutoring and Learning Centre, George Brown College 2014

www.georgebrown.ca/tlc

Adding and Subtracting Mixed Numbers and Improper Fractions

Exercises: 1. Add or subtract the following improper fractions and mixed numbers. Remember to reduce where possible.

2. Each week Fred works 3Β½ hours on Monday, 3 hours on Tuesday, 2 hours on Wednesday, 2ΒΌ hours on Thursday, and 4 hours on Friday. How many hours does he work per week?

3. During a workshop, the English Tutors ate 3Β½ pizzas and the Math Tutors ate 5β pizzas. How many pizzas were ordered? (Hint: Pizzas are ordered in wholes.)

4. The fourth floor of the D building has 600Β½ ft2 of space to house the TLC (Tutoring Learning Centre), SLC (Student Learning Centre), and PAL (Peer Assisted Learning). If the TLC uses 120ΒΌ ft2 and the PAL uses 115β ft2, how much space does SLC use?

5. It takes 2β hours to travel to Toronto from Waterloo while travelling with the GO. However, driving takes 1β hours. How much time do you save by driving?

Solutions: 1.

2.

3. 10 pizzas

4.

Tutoring and Learning Centre, George Brown College 2014

5. www.georgebrown.ca/tlc

Just like our counting numbers (1, 2, 3,β¦), fractions can also be added and subtracted. When counting improper fractions and mixed numbers, we are counting the number wholes and parts.

Note: The rules for adding and subtracting improper fractions are the same as working with proper fractions.

Case 1: Adding and Subtracting Improper Fractions with Common Denominators Step 1: Keep the denominator the same. Step 2: Add or subtract the numerators. Step 3: If the answer is an improper form, reduce the fraction into a mixed number.

Exercise 1: Add the fractions, + . Letβs draw a picture to see what this looks like.

The 4 in the denominator tells us that each whole is cut into 4 equal portions. By adding the fractions we are grouping the total number of wholes and parts.

We have 5 slices and each whole is made up of 4 slices, .

We have 6 slices and each whole is made up of 4 slices, .

Altogether, we have 2 wholes and 3 quarters,

How does the math work?

Step 1: Since the two fractions have equal sized slices, keep the denominator the same, .

Step 2: Add the numerators,

.

Step 3: Thus, we have wholes. Tutoring and Learning Centre, George Brown College 2014

www.georgebrown.ca/tlc

Adding and Subtracting Mixed Numbers and Improper Fractions

Case 2: Adding and Subtracting Improper Fractions with Different Denominators Step 1: Find the Lowest Common Multiple (LCM) between the denominators. Step 2: Multiply the numerator and denominator of each fraction by a number so that they have the LCM as their new denominator. Step 3: Add or subtract the numerators and keep the denominator the same. Step 4: If the answer is an improper form, reduce the fraction into a mixed number.

Exercise 2: Subtract the fractions,

.

Step 1: List the multiples of 6 and 8.

Multiplies of 6: 6, 12, 18, 24, 30, 36, 48β¦ Multiplies of 8: 8, 16, 24, 32, 40, 48, 56β¦

The Lowest Common Multiple between 6 and 8 is 24.

Step 2: a) We need to find a number that when multiplied to the top and bottom of , we get the LCM (24) as the new denominator.

Since

, we need to multiply the numerator and the denominator by 4.

Thus, is equivalent to

b) We need to find a number that when multiplied to the top and bottom of , we get the LCM (24) as the new denominator.

Since

, we need to multiply the numerator and the denominator by 3.

Thus, is equivalent to

Step 3: Since our fractions now have equal sized slices, we can subtract their

numerators. Thus, we now have,

of a whole.

Tutoring and Learning Centre, George Brown College 2014

www.georgebrown.ca/tlc

Adding and Subtracting Mixed Numbers and Improper Fractions

Case 3: Adding and Subtracting Mixed Numbers Method 1 Step 1: Convert all mixed numbers into improper fractions. Step 2: Check! Do they have a common denominator? If not, find a common denominator. Step 3: When necessary, create equivalent fractions. Step 4: Add or subtract the numerators and keep the denominator the same. Step 5: If the answer is an improper form, reduce the fraction into a mixed number.

Exercise 3: Subtract the fractions,

.

Step1: Convert both mixed numbers into improper fractions.

π

ππ

ππ π

π

π

ππ π

Step 2: List the multiples of 4 and 7.

Multiplies of 4: 4, 8, 12, 16, 20, 24, 28β¦ Multiplies of 7: 7, 14, 21, 28, 35β¦

The Lowest Common Multiple between 4 and 7 is 28.

Step 3: a) We need to find a number that when multiplied to the top and bottom of , we get the LCM (28) as the new denominator.

Since

, we need to multiply the numerator and the denominator by 7.

Thus, is equivalent to

b) We need to find a number that when multiplied to the top and bottom of , we get the LCM (28) as the new denominator.

Tutoring and Learning Centre, George Brown College 2014

www.georgebrown.ca/tlc

Adding and Subtracting Mixed Numbers and Improper Fractions

Since

, we need to multiply the numerator and the denominator by 4.

Thus, is equivalent to

Step 4: Since our fractions now have equal sized slices, we can subtract their

numerators. Subtracting their numerators we have,

of a whole.

Step 5: Thus, we have wholes.

Case 4: Adding and Subtracting Mixed Numbers Method 2 In this second method, we will break the mixed number into wholes and parts.

Step 1: Add or subtract the whole number part. Step 2: Check! Does the fraction part share a common denominator? If not, find one. Step 3: When necessary, create equivalent fractions. Step 4: Add or subtract the numerators of the fraction part and keep the denominator the same. Step 5: If the answer is an improper form, reduce the fraction into a mixed number.

Exercise 4: Jessica is years old today. How old was she years ago?

Since we are looking at the difference between her current and past ages, our equation

will look like,

.

Step 1: Subtract the whole number part,

Step 2: List the multiples of 2 and 4. The Lowest Common Multiple between 2 and 4 is 4.

Multiplies of 2: 2, 4, 6, 8β¦ Multiplies of 4: 4, 8, 12β¦

Step 3: a) We need to find a number that when multiplied to the top and bottom of , we get the LCM (4) as the new denominator.

Since

, we need to multiply the numerator and the denominator by 2.

Tutoring and Learning Centre, George Brown College 2014

www.georgebrown.ca/tlc

Adding and Subtracting Mixed Numbers and Improper Fractions

Thus, is equivalent to

b) Since already has the LCM (4) as the denominator, we leave the fraction as it is.

Step 4: Since our fraction part now has equal sized slices, we can subtract their

numerators. Subtracting their numerators we have,

of a whole.

Step 5: Combining our whole number and fraction parts we get,

.

Exercise 5: Subtract the fractions,

Step 1: Subtracting the whole number part, we get Step 2: Subtracting the fraction part, we get

of a whole.

Since we cannot take 3 away from 1, we need to borrow a whole from the first fraction.

Given

, letβs borrow a whole by following the steps below: Rewrite 3 wholes into 2 wholes + 1 whole.

Since each whole has 4 slices, add the four slices from the borrowed whole into the numerator of the fraction part.

Thus, we have created an equivalent fraction where =

Step 3: Now we are able to subtract the fractions,

.

Subtracting the whole number part, we are left with,

Subtracting the fraction part, we are left with,

= of a whole.

Combining our whole number and fraction parts we are left with,

. Tutoring and Learning Centre, George Brown College 2014

www.georgebrown.ca/tlc

Adding and Subtracting Mixed Numbers and Improper Fractions

Exercises: 1. Add or subtract the following improper fractions and mixed numbers. Remember to reduce where possible.

2. Each week Fred works 3Β½ hours on Monday, 3 hours on Tuesday, 2 hours on Wednesday, 2ΒΌ hours on Thursday, and 4 hours on Friday. How many hours does he work per week?

3. During a workshop, the English Tutors ate 3Β½ pizzas and the Math Tutors ate 5β pizzas. How many pizzas were ordered? (Hint: Pizzas are ordered in wholes.)

4. The fourth floor of the D building has 600Β½ ft2 of space to house the TLC (Tutoring Learning Centre), SLC (Student Learning Centre), and PAL (Peer Assisted Learning). If the TLC uses 120ΒΌ ft2 and the PAL uses 115β ft2, how much space does SLC use?

5. It takes 2β hours to travel to Toronto from Waterloo while travelling with the GO. However, driving takes 1β hours. How much time do you save by driving?

Solutions: 1.

2.

3. 10 pizzas

4.

Tutoring and Learning Centre, George Brown College 2014

5. www.georgebrown.ca/tlc

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