The Three Steps

Convert $2\frac{3}{4}$ to an improper fraction.

Multiply the whole number by the denominator: $2 \times 4 = 8$ = $8$ fourths in the $2$ wholes

Add the numerator: $8 + 3 = 11$ = $11$ fourths in total

Keep the denominator: The pieces are still fourths = $\frac{11}{4}$

Answer: $2\frac{3}{4} = \frac{11}{4}$

Seeing It with Pieces

Draw $1\frac{2}{3}$ with fraction bars and count the thirds.

Cut the whole bar into thirds: $1 = \frac{3}{3}$ = $3$ thirds

Add the extra thirds: $3 + 2 = 5$ = $5$ thirds

Write the fraction: Five pieces, each one third = $\frac{5}{3}$

Answer: $1\frac{2}{3} = \frac{5}{3}$

Converting and Checking

Convert $3\frac{2}{5}$ to an improper fraction and check your answer.

Multiply the whole number by the denominator: $3 \times 5 = 15$ = $15$ fifths

Add the numerator: $15 + 2 = 17$ = $17$ fifths

Keep the denominator: $\frac{17}{5}$ = $\frac{17}{5}$

Check by dividing back: $17 \div 5 = 3$ remainder $2$ = $3\frac{2}{5}$, correct!

Answer: $3\frac{2}{5} = \frac{17}{5}$

A Whole Number as a Fraction

Write the whole number $3$ as a fraction with denominator $4$.

Count the fourths in one whole: $1 = \frac{4}{4}$ = $4$ fourths

Multiply by the number of wholes: $3 \times 4 = 12$ = $12$ fourths

There is no fraction part to add: $12 + 0 = 12$ = $\frac{12}{4}$

Answer: $3 = \frac{12}{4}$ (and also $3 = \frac{3}{1}$)

Bigger Numbers, Same Rule

Convert $6\frac{7}{12}$ to an improper fraction.

Multiply the whole number by the denominator: $6 \times 12 = 72$ = $72$ twelfths

Add the numerator: $72 + 7 = 79$ = $79$ twelfths

Keep the denominator: $\frac{79}{12}$ = $\frac{79}{12}$

Check by dividing back: $79 \div 12 = 6$ remainder $7$ = $6\frac{7}{12}$, correct!

Answer: $6\frac{7}{12} = \frac{79}{12}$

Mistake: Adding the whole number to the numerator: $2\frac{3}{4} \rightarrow \frac{2 + 3}{4} = \frac{5}{4}$

Why: Each whole is worth $4$ fourths, not $1$. Two wholes are $8$ fourths, so just adding $2$ loses $6$ pieces.

Correct: Multiply first, then add: $\frac{2 \times 4 + 3}{4} = \frac{11}{4}$

Mistake: Multiplying the whole number by the numerator: $2\frac{3}{4} \rightarrow \frac{2 \times 3 + 4}{4} = \frac{10}{4}$

Why: The denominator tells how many pieces make one whole, so it is the number you multiply by.

Correct: Whole number times **denominator**: $2 \times 4 = 8$, then $8 + 3 = 11$, so $\frac{11}{4}$

Mistake: Changing the denominator: $2\frac{3}{4} \rightarrow \frac{11}{8}$

Why: The size of the pieces does not change when you count them differently. They are still fourths.

Correct: Keep the denominator: $2\frac{3}{4} = \frac{11}{4}$

Mistake: Forgetting to add the numerator: $2\frac{3}{4} \rightarrow \frac{8}{4}$

Why: $\frac{8}{4}$ is only the $2$ wholes. The $3$ extra fourths are missing.

Correct: Always finish with the add step: $8 + 3 = 11$, so $\frac{11}{4}$

Measuring in the Kitchen

Recipes use mixed numbers, but measuring cups come in fixed sizes. Converting tells you how many scoops you need.

$1\frac{3}{4}$ cups of sugar is $\frac{7}{4}$ cups, so $7$ scoops with a quarter cup measure.

Counting Pizza Slices

At a party, leftover pizzas are often described as mixed numbers. The improper fraction tells you the number of slices.

$1\frac{5}{6}$ pizzas cut into sixths are $\frac{11}{6}$, that is $11$ slices.

Time in Quarter Hours

A quarter of an hour is $15$ minutes. Converting a time like $1\frac{3}{4}$ hours to quarters makes it easy to find the minutes.

$2\frac{1}{4}$ hours is $\frac{9}{4}$ hours, that is $9$ quarter hours or $135$ minutes.

Q: Why do we keep the same denominator?

A: The denominator tells the size of the pieces. When you convert $2\frac{3}{4}$, you only count the pieces again, and they are still fourths. That is why the answer is $\frac{11}{4}$, not $\frac{11}{8}$.

Q: How do I write a whole number as a fraction?

A: Put it over $1$: $3 = \frac{3}{1}$. If you need another denominator, multiply: $3 = \frac{3 \times 4}{4} = \frac{12}{4}$. This is the same rule with a fraction part of $0$.

Q: How can I check my answer?

A: Divide the numerator by the denominator. For $\frac{17}{5}$: $17 \div 5 = 3$ remainder $2$, so you get back $3\frac{2}{5}$. This reverse step is the lesson Converting Improper Fractions to Mixed Numbers.

Q: Why convert at all if the mixed number is easier to read?

A: Mixed numbers are great for reading amounts, but improper fractions are better for calculating. When you multiply or divide mixed numbers in later lessons, you will always convert them to improper fractions first.