KS3 · Maths

Converting between improper fractions and mixed numbers

247 people, and each bus seats 80. That's 247/80 busloads. So how many buses do you book? Right now it's hard to tell. Soon it'll be obvious.

Count the fifths

Where does twelve fifths land?

This line runs from 0 to 3, and every whole is split into five equal parts: fifths. Each nudge of a slider moves exactly one fifth. Place all five values, then check. Watch where 12/5 and 2 2/5 end up.

Name the fraction

Less than one, exactly one, or more?

Pick a fraction, then pick where it belongs.

Is it less than one, exactly one, or greater than one?

Still to sort

Less than one (0)

The numerator is smaller than the denominator.

Where the line is: Push the numerator up until it matches the denominator and you land exactly on 1, not past it.

Exactly one (0)

The numerator and the denominator are the same.

Where the line is: This is the exact spot where proper stops and improper begins.

Greater than one (0)

The numerator is bigger than the denominator.

6 of 6 still to sort.

Sort each fraction by its size first. Then switch the rule and sort the same fractions by name. Watch which ones move together.

Watch out: An improper fraction is greater than OR EQUAL TO one. 'An improper fraction is always greater than one' is false: 5/5 is improper, and it equals exactly 1.

Read it properly

What does 3 1/2 actually mean?

A mixed number is a whole number with a proper fraction written next to it, like 3 1/2. It's how you'd write an improper fraction once the wholes have been counted out.

Which is closest to what you think 3 1/2 means?
How sure are you?

Improper to mixed, the fast way

Fill in the missing steps

Write 22/7 as a mixed number. Then do the same for 46/4.

  1. Read 22/7 as 22 ÷ 7. The numerator, 22, is the number being divided. The denominator, 7, is what you divide by.Why divide? 22 ÷ 7 asks how many groups of seven sevenths, which means how many wholes, fit into 22 sevenths. It's the counting you did on the number line, done in one go.
  2. missing step
Which line is step 2?

Going backwards: mixed to improper

Problem

Write 8 1/2 and 5 6/7 as improper fractions.

Predict, then check

Decide before you convert anything.

Which is greater: 17 3/4 or 35/2?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Twelve fifths, or two and two fifths? Same amount, two ways to write it. Learn to switch between them, and to pick the one that helps.

What you need to know

  • Proper fraction: numerator less than denominator, so less than one (for example 3/8).
  • Improper fraction: numerator greater than or equal to denominator, so greater than or equal to one (for example 7/5 or 5/5).
  • Mixed number: a whole number plus a proper fraction (for example 3 1/2 = 3 + 1/2).
  • Improper to mixed: divide top by bottom. Mixed to improper: turn the wholes into fractions with the same denominator, then add.

The big picture

A proper fraction has a smaller numerator than denominator, so it's less than one. An improper fraction has a numerator greater than or equal to its denominator, so it's at least one. A mixed number, like 2 2/5, is a whole number plus a proper fraction, and it names the same amount as an improper fraction, like 12/5. To convert, divide the numerator by the denominator; to go back, write the wholes with the same denominator and add. Then use whichever form suits the job.

Key points

1The denominator tells you how many equal parts make one whole.
2An improper fraction and its mixed number are the same amount written two ways: 12/5 = 2 2/5.
3An improper fraction can equal one (9/9 = 1), so it isn't always greater than one.
4When you divide, the quotient becomes the whole number, the remainder becomes the numerator and the denominator stays the same.
5You can simplify before or after converting: 15/6 = 2 1/2 either way.
6Mixed numbers make size clear in real contexts; improper fractions are often easier to calculate with.

Worked example

Problem

247 people are going on a trip, and each bus seats 80. How many buses do you need to book?

⚠ Watch out

Swapping the whole number and the remainder. For 14/3, 3 goes into 14 four times with 2 left over, so the answer is 4 2/3, not 2 4/3. A quick sense check: the fraction part of a mixed number must be proper, and 4/3 isn't.

🧠

Memory hook

Picture pizzas cut into slices. A mixed number counts the whole pizzas first, then the spare slices. An improper fraction just counts every slice. Same pizza, two ways of counting it.

✓

Check yourself

Write 50/3 as a mixed number, then turn 7 2/9 into an improper fraction. (Answers: 16 2/3 and 65/9.)

Flashcards

(14)
What is a proper fraction?
A fraction whose numerator is less than its denominator, such as 3/8. It is always less than one.
What is an improper fraction?
A fraction whose numerator is greater than or equal to its denominator, such as 7/5 or 5/5. It is always greater than or equal to one.
True or false: an improper fraction is always greater than one.
False. It can equal one: 9/9 is improper and equals 1.
What is a mixed number?
A whole number plus a proper fraction, such as 3 1/2, which means 3 + 1/2.
What does the denominator tell you?
How many equal parts make one whole. In 12/5, five fifths make each whole.
How do you turn an improper fraction into a mixed number by dividing?
Divide the numerator by the denominator. The quotient is the whole number, the remainder is the new numerator, and the denominator stays the same.
Write 13/4 as a mixed number.
3 1/4 (4 goes into 13 three times, with 1 left over).
How do you write a whole number as an improper fraction?
Multiply it by n/n for the denominator you want. For example, 3 = 3 × 4/4 = 12/4.
How many improper fractions are equal to 4?
Infinitely many: 8/2, 12/3, 16/4, 40/10, 80/20 and so on.
How do you turn a mixed number into an improper fraction?
Write the whole number as a fraction with the same denominator as the fraction part, then add. For example, 4 2/3 = 12/3 + 2/3 = 14/3.
Write 11 3/5 as an improper fraction.
58/5 (11 = 55/5, and 55/5 + 3/5 = 58/5).
Should you simplify before or after converting 142/4?
Either works. 142/4 = 71/2 = 35 1/2, and dividing first gives 35 2/4, which simplifies to the same 35 1/2.
Which is greater, 24/5 or 5 1/5?
5 1/5. Written in fifths it is 26/5, which is more than 24/5.
Which form makes size clear in a real situation, and which is often easier for adding?
A mixed number shows how many wholes there are at a glance. Improper fractions are often easier to add.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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