A complex fraction is a fraction that has another fraction in its numerator (top part) or denominator (bottom part). For example, the fraction 3/4 is not a complex fraction, but the fraction (3/4)/(5/6) is a complex fraction because it has a fraction, 3/4, in the numerator.

In a word problem involving complex fractions, you might be asked to perform arithmetic operations on fractions and simplify the result. For example, you might be asked to add, subtract, multiply, or divide complex fractions, or to simplify a complex fraction by reducing it to lowest terms.

Here is an example of a word problem involving complex fractions:

Word problem: A recipe calls for 3/8 cup of sugar and 1/4 cup of flour. How much sugar and flour do you need in total?

Solution: To find the total amount of sugar and flour, we need to add the fractions 3/8 and 1/4. We can do this by finding the least common multiple of the two denominators, 8 and 4, which is 8. We can then rewrite the fractions with a denominator of 8:

3/8 + 1/4 = (32)/(82) + (12)/(42) = 6/16 + 2/8 = (6+2)/8 = 8/8 = 1

So the total amount of sugar and flour needed is 1 cup.

## Practice

Here is a sample word problem that involves complex fractions:

Word problem: The school store sells pens in packs of 8 and folders in packs of 5. If a customer wants to buy an equal number of pens and folders, and the total cost of the pens and folders is \$24, what is the cost of each pen and folder?

Solution: Let’s say that the cost of a pen is p dollars and the cost of a folder is f dollars. We can set up the following equation to represent the problem:

(8p + 5f) = 24

To solve this equation, we need to get the variables on one side of the equal sign and the constants on the other side. We can do this by dividing both sides of the equation by 13, which is the least common multiple of 8 and 5:

(8p/13 + 5f/13) = 24/13

This simplifies to:

(4/13)p + (5/13)f = 24/13

To solve for p, we can subtract (5/13)f from both sides of the equation:

(4/13)p = 24/13 – (5/13)f

This simplifies to:

(4/13)p = (24 – 5f)/13

To solve for p, we can multiply both sides of the equation by 13/4:

p = (24 – 5f)*(13/4)

This simplifies to:

p = (3 – (5/4)f)

Now that we have an equation for p, we can solve for f in a similar way. We can start by rearranging the equation above to get p by itself on one side:

p – (3 – (5/4)f) = 0

This simplifies to:

p = 3 – (5/4)f

We can substitute this expression for p in the original equation:

(8(3 – (5/4)f) + 5f) = 24

This simplifies to:

(24 – (5/4)f + 5f) = 24

We can combine like terms:

(29/4)f = 24

To solve for f, we can divide both sides of the equation by (29/4):

f = 24/(29/4)

This simplifies to:

f = 8/29

Now we have the cost of both pens and folders:

• The cost of a pen is p = \$3.
• The cost of a folder is f = \$8/29.

## More Examples

Here are five word problems involving complex fractions:

### Questions

1. Word problem: The school store sells pens in packs of 8 and folders in packs of 5. If a customer wants to buy an equal number of pens and folders, and the total cost of the pens and folders is \$24, what is the cost of each pen and folder?
2. Word problem: A recipe calls for 3/8 cup of sugar and 1/4 cup of flour. How much sugar and flour do you need in total?
3. Word problem: A carpenter has a board that is 8 feet long and needs to cut it into pieces that are 3/4 foot long. How many pieces can the carpenter cut from the board?
4. Word problem: A jar contains 1/3 pound of jellybeans. If a person eats 1/4 pound of jellybeans, how many pounds of jellybeans are left in the jar?
5. Word problem: A student has a test with 25 questions and gets 22 of them correct. What fraction of the questions did the student answer correctly?

### Answers

1. Word problem: The school store sells pens in packs of 8 and folders in packs of 5. If a customer wants to buy an equal number of pens and folders, and the total cost of the pens and folders is \$24, what is the cost of each pen and folder?

Solution: Let’s say that the cost of a pen is p dollars and the cost of a folder is f dollars. We can set up the following equation to represent the problem:

(8p + 5f) = 24

To solve this equation, we need to get the variables on one side of the equal sign and the constants on the other side. We can do this by dividing both sides of the equation by 13, which is the least common multiple of 8 and 5:

(8p/13 + 5f/13) = 24/13

This simplifies to:

(4/13)p + (5/13)f = 24/13

To solve for p, we can subtract (5/13)f from both sides of the equation:

(4/13)p = 24/13 – (5/13)f

This simplifies to:

(4/13)p = (24 – 5f)/13

To solve for p, we can multiply both sides of the equation by 13/4:

p = (24 – 5f)*(13/4)

This simplifies to:

p = (3 – (5/4)f)

Now that we have an equation for p, we can solve for f in a similar way. We can start by rearranging the equation above to get p by itself on one side:

p – (3 – (5/4)f) = 0

This simplifies to:

p = 3 – (5/4)f

We can substitute this expression for p in the original equation:

(8(3 – (5/4)f) + 5f) = 24

This simplifies to:

(24 – (5/4)f + 5f) = 24

We can combine like terms:

(29/4)f = 24

To solve for f, we can divide both sides of the equation by (29/4):

f = 24/(29/4)

This simplifies to:

f = 8/29

Now we have the cost of both pens and folders:

• The cost of a pen is p = \$3.
• The cost of a folder is f = \$8/29.
1. Word problem: A recipe calls for 3/8 cup of sugar and 1/4 cup of flour. How much sugar and flour do you need in total?

Solution: To find the total amount of sugar and flour, we need to add the fractions 3/8 and 1/4. We can do this by finding the least common multiple of the two denominators, 8 and 4, which is 8. We can then rewrite the fractions with a denominator of 8:

3/8 + 1/4 = (32)/(82) + (12)/(42) = 6/16 + 2/8 = (6+2)/8 = 8/8 = 1

So the total amount of sugar and flour needed is 1 cup.

1. Word problem: A carpenter has a board that is 8 feet long and needs to cut it into pieces that are 3/4 foot long. How many pieces can the carpenter cut from the board?

Solution: To find the number of pieces that the carpenter can cut from the board, we need to divide the length of the board by the length of each piece. We can do this by dividing the numerator of the fraction representing the length of the board by the numerator of the fraction representing the length of each piece:

8/3 = 8/3/4/4 = 8/34/4 = 32/12 = 32/12/4/4 = 32/43/3 = 8

So the carpenter can cut 8 pieces from the board.

1. Word problem: A jar contains 1/3 pound of jellybeans. If a person eats 1/4 pound of jellybeans, how many pounds of jellybeans are left in the jar?

Solution: To find the number of pounds of jellybeans left in the jar, we need to subtract the fraction representing the amount eaten from the fraction representing the total amount. We can do this by finding the least common multiple of the two denominators, 3 and 4, which is 12. We can then rewrite the fractions with a denominator of 12:

1/3 – 1/4 = (14)/(34) – (13)/(43) = 4/12 – 3/12 = (4-3)/12 = 1/12

So there is 1/12 pound of jellybeans left in the jar.

1. Word problem: A student has a test with 25 questions and gets 22 of them correct. What fraction of the questions did the student answer correctly?

Solution: To find the fraction of questions that the student answered correctly, we can divide the number of questions the student got correct by the total number of questions. We can represent this as a fraction by putting the number of correct answers over the total number of questions:

22/25 = 22/25/1/1 = 22/125/25 = 88/25 = 88/25/5/5 = 88/55/5 = 17/5

So the student answered 17/5 of the questions correctly.