Fractions bigger than one
Tonight's idea
Cut some pizzas into quarters and take seven pieces. You have seven quarters, written \(\tfrac74\). The top is bigger than the bottom. It means you have more than one whole.
Four of the quarters make one whole pizza. Three are left over. So \(\tfrac74\) is "one and three quarters", written \(1\tfrac34\). A whole number with a fraction beside it is called a mixed number.
A fraction with the top bigger than the bottom is called an improper fraction. The name is unfair: nothing is wrong with it.
To make a mixed number, divide the top by the bottom. The answer is the wholes. The remainder is the pieces left over.
Here \(7 \div 4 = 1\) remainder 3: one whole, three quarters left.
To go back, count the pieces. Each whole is 4 quarters. \(2\tfrac14\) is \(2 \times 4 = 8\) quarters and 1 more: \(\tfrac94\).
Watch one: \(\tfrac{17}{5}\) and back
How many wholes are in 17 fifths? Five fifths make a whole, so divide: \(17 \div 5 = 3\) remainder 2. That is 3 wholes and 2 fifths.
\[\tfrac{17}{5} = 3\tfrac25\]
On the number line it sits between 3 and 4, a little before half way.
Check by going back. Three wholes are \(3 \times 5 = 15\) fifths. Add the 2 extra: \(15 + 2 = 17\) fifths.
Your turn
- Write as a mixed number: (a) \(\tfrac52\); (b) \(\tfrac{11}{4}\); (c) \(\tfrac{14}{3}\).
- Write as an improper fraction: (a) \(1\tfrac23\); (b) \(3\tfrac12\); (c) \(2\tfrac58\).
- This number line is marked in quarters. Mark and label \(\tfrac12\), \(\tfrac54\), \(2\tfrac34\) and \(\tfrac72\).
- Five children each get \(\tfrac34\) of a pizza. (a) How many quarters is that in all? (b) Write it as a mixed number. (c) Pizzas are only sold whole. How many must you buy, and how much is left over?
Stretch: half way
What number is exactly half way between \(1\tfrac12\) and \(2\tfrac14\)? Give it as a mixed number.
Answers
- (a) \(2\tfrac12\). (b) \(2\tfrac34\). (c) \(4\tfrac23\), since \(14 \div 3 = 4\) remainder 2.
- (a) \(\tfrac53\). (b) \(\tfrac72\). (c) \(\tfrac{21}{8}\), since \(2 \times 8 + 5 = 21\).
- Count ticks from 0. \(\tfrac12\) is the 2nd tick. \(\tfrac54\) is the 5th, one past 1. \(2\tfrac34\) is the 11th, one before 3. \(\tfrac72 = 3\tfrac12\) is the 14th, half way between 3 and 4.
- (a) \(5 \times 3 = 15\) quarters. (b) \(\tfrac{15}{4} = 3\tfrac34\). (c) Buy 4 pizzas. That is 16 quarters, so 1 quarter is left over.
Stretch. \(1\tfrac78\). In quarters the two numbers are \(\tfrac64\) and \(\tfrac94\). In eighths they are \(\tfrac{12}{8}\) and \(\tfrac{18}{8}\). Half way between 12 and 18 is 15, and \(\tfrac{15}{8} = 1\tfrac78\).