Succeed with maths – Part 1
Succeed with maths – Part 1

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Succeed with maths – Part 1

4.1 Improper fractions

Moving on from proper fractions, where the numerator is less than the denominator, you come to improper fractions.

These are fractions in which the numerator is greater than or equal to the denominator. You can think of these as ‘top-heavy’ fractions, such as five divided by two and eight divided by five. These mean 5 halves, and 8 fifths respectively.

Improper fractions are another way of showing mixed numbers. So, can be rewritten as an improper fraction to give seven divided by three. You can again imagine this in terms of pizzas. If you have two and one third pizzas, and cut each of the whole pizzas into thirds, then how many thirds of a pizza will you have?

Described image
Figure 13 Seven thirds

In this case, the total number of thirds will be open two multiplication three close plus one equals seven, showing that can indeed be written as seven divided by three.

Now think about how seven thirds was found. Because there were two whole parts broken into thirds, plus one extra third, 2 was multiplied by 3 to calculate how many thirds there were in two whole pizzas, then 1 was added. Since these are all thirds, the 7 was placed over the 3 in fractional notation. This can be worked out as follows:

equation sequence open two multiplication three close plus one divided by three equals six plus one divided by three equals seven divided by three

This works for any mixed number that you need to convert into an improper fraction. So, the rule is:

mixed number equals left parenthesis whole multiplication denominator right parenthesis plus numerator divided by denominator equals improper fraction

You can also change improper fractions back into mixed numbers. For example, for 17 divided by eight imagine some pizzas have been cut into eight equal slices (eighths), and that you have 17 slices but you don’t know how many whole pizzas this makes. You know that eight slices make one whole pizza, and that two pizzas would be 16 slices (2 x 8). There would be one-eighth (one slice) left over. So, 17 divided by eight equals. You can also carry out the division implied by the fraction:

multiline equation line 1 multiline equation line 1 two line 2 equation sequence 17 divided by eight equals 17 division eight equals eight times 17 line 2 minus minus 16 low line line 3 one

Since eight goes into 17 at most two whole times, and there is one out of eight parts left over, this again gives 17 divided by eight equals.

One of the best ways for you to cement new ideas in your mind is to practice them and you’ll get this chance in the next activity for mixed numbers and improper fractions.

Activity 6 Mixed numbers and improper fractions

Timing: Allow approximately 10 minutes

(a) Change the following mixed numbers into improper fractions.

  • i.

  • ii.

  • iii.

Comment

If you are having trouble with this, did you try using a picture?

Answer

(a)

(i) You are working in quarters, so using ‘pizza maths’ you need to divide each pizza into four to give you this:

As there are four quarters in each whole, three wholes will give 3 × 4 = 12 quarters. The one extra quarter makes 13 quarters overall.

Thus, equals 13 divided by four.

(ii) You could use the following shortcut to solve this one:

equation sequence left parenthesis whole multiplication denominator right parenthesis plus numerator divided by denominator equals open two multiplication five close plus four divided by five equals 10 plus four divided by five equals 14 divided by five

(iii) There are eight eighths in each whole, so seven wholes will give 56 eighths (7 x 8). The extra three eighths makes 59 overall, so the fraction is 59 divided by eight. Hence, equals 59 divided by eight.

(b) Change the following improper fractions into mixed numbers.

  • i.23 divided by five

  • ii.15 divided by seven

  • iii.17 divided by four

Answer

(b)

  • i.Since 4 × 5 = 20, 20 fifths will make up 4 wholes. This leaves an extra 3 fifths, so the fraction is . Thus, 23 divided by five equals.

  • ii.Let’s try using long division:

    multiline equation line 1 multiline equation line 1 two line 2 seven times 15 line 2 minus minus 14 low line line 3 one

    Since 2 × 7 = 14, 14 sevenths will make up two wholes. This leaves one seventh over, so the fraction is . Hence, 15 divided by seven equals.

  • iii.Since 4 × 4 = 16, 16 quarters will make up four wholes. This leaves one quarter over, so the fraction is . Consequently, 17 divided by four equals.

Well done for completing this activity with mixed numbers and improper fractions. You will need the skills that you have been practising here next week when you learn how to carry out calculations with fractions. You’ve got one last activity in the next section before finishing.

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