Fraction relationships in bar models

Let the bar stand for a named whole

A bar model shows relationships between quantities. Its equal parts stand for equal amounts; the drawn length is not a measurement you need to make with a ruler.

A jug starts full. One third of its water is used. Draw a bar for the original amount, split into three equal parts. One part is used and two remain. Both fractions refer to that same original amount.

Try the model. You can change your answer or look at the worked model. This activity has no score.

Check one original whole, equal thirds, and how the used and remaining amounts fit together. You can also draw on paper and compare with the public answer.

Show solution

Used part: 1/3 whole. Reference whole: 1 whole. Bars use the same scale.

1/3

Remaining part: 2/3 whole. Reference whole: 1 whole. Bars use the same scale.

1/31/3

Original whole: 1 whole. Reference whole: 1 whole. Bars use the same scale.

1/31/31/3

In the original-whole box, show exactly three thirds. Doubling every box keeps the same ratio but describes two original wholes.

Name each unit before writing a calculation: one block is one third of the original amount. Two remaining blocks are two thirds of the original amount. Half of the remainder is one of those blocks, so it is one third of the original amount.

Work backwards from a known part

Aisha uses 2/5 of a ribbon. She has 18 cm left. How long was the ribbon originally?

The whole bar is the original ribbon. Two of its five equal parts were used; three parts remain. Therefore three parts represent 18 cm.

One part represents 18÷3 = 6 cm. The original five parts represent 5 × 6 = 30 cm. Check: two parts used are 12 cm, and 12 + 18 = 30 cm.

An equally valid calculation is 18 × 5÷3 = 30 cm. It scales three equal parts up to five equal parts. The bar explains why the known 18 cm belongs to three parts.

A tempting calculation is 2/5 × 18. That finds two fifths of the remainder. The question says two fifths of the original ribbon, so it describes a different amount.

Check your understanding 1 A fraction of what remains

Ben starts with 40 beads. He gives away 1/4 of them. He then uses 2/3 of the beads remaining. How many beads does he use at the second step?

Beads used at the second step

Check answer

Show answer

He first gives away 1/4 × 40 = 10 beads. The new reference whole is the 30 remaining beads. He uses 2/3 × 30 = 20 beads. Finding two thirds of 40 would use the wrong whole. Ten beads remain at the end, which checks 10 + 20 + 10 = 40.

Link two changes without changing the original whole

Kai gives away 1/4 of his postcards. He then gives away 1/3 of the remaining postcards. He has 24 postcards left at the end. How many postcards did he have originally?

Sketch your own bars first. Label the original collection, the first gift, the first remainder, the second gift and the final remainder. Which amount does the second fraction refer to?

Your bars or calculations, and the whole used for each fraction

Try the model. You can change your answer or look at the worked model. This activity has no score.

Check one original collection, both part–whole relationships, and the fraction used at each stage. The second third must refer to the first remainder. You can also draw on paper and compare with the public answer.

Show solution

Original collection: 1 original collection. Reference whole: 1 original collection. Bars use the same scale.

1/41/41/41/4

Given away first: 1/4 original collection. Reference whole: 1 original collection. Bars use the same scale.

1/4

Remaining after the first gift: 3/4 original collection. Reference whole: 1 original collection. Bars use the same scale.

1/41/41/4

Given away second: 1/4 original collection. Reference whole: 1 original collection. Bars use the same scale.

1/4

Left at the end: 1/2 original collection. Reference whole: 1 original collection. Bars use the same scale.

1/41/4

One quarter-block always represents one quarter of the original collection. At the second stage, that same block also represents one third of the remainder. The amount has two fraction names because the named whole has changed.

The rows are separate views of the same postcards, not five collections to add together. In your sketch, you can put the two gifts and the final remainder along one original bar instead.

Check your understanding 2 Turn the model into a count

The final two original parts represent 24 postcards. Find the original count. Use the model or work backwards; explain your choice on paper.

Original number of postcards

Check answer

Show answer

Equal original parts: two parts are 24 postcards, so one part is 24÷2 = 12 postcards. Four parts are 4 × 12 = 48 postcards.

The fraction product describes the second gift:

1/3 × 3/4 = 1/4.

It is one third of the first remainder, which was three quarters of the original. The final fraction is 3/4-1/4 = 1/2 of the original. Half is 24 postcards, so the original count is 48.

Work backwards through the two remainders: the final 24 postcards are two thirds of the first remainder. The first remainder is 24÷2 × 3 = 36 postcards. This is three quarters of the original, so the original count is 36÷3 × 4 = 48 postcards.

Both methods explain the same relationships. Check forwards: the first gift is 12 postcards, leaving 36. The second gift is 36÷3 = 12 postcards, leaving 24. The gifts and final remainder satisfy 12 + 12 + 24 = 48.

Do not add 1/4 + 1/3 to find the fraction given away: those fractions name different wholes. First express the second gift as a fraction of the original collection. Also keep a relationship error separate from a calculation slip. If your model correctly shows two final parts but you divide 24 by four, revisit which parts the known 24 labels.

Fade the drawing when the relationship is clear

A tank contains some water. After 3/8 of it is used, 25 litres remain. Complete the relationship before calculating:

Remaining amount = remaining fraction × original amount.

The remaining fraction, one-part amount and original volume

Transfer the relationship to a different unknown

Check your understanding 3 Two purchases, one original amount

Mei spends 1/4 of her original money on a book and 3/8 of her original money on a bag. The bag costs 6 dollars more than the book. How much money did she have originally?

Your model or equations and conclusion

Show answer

Use eighths of the original amount. The book costs two eighths and the bag costs three eighths. Their difference is one eighth, worth 6 dollars. The original eight eighths were worth 48 dollars. Check: the book cost 12 dollars and the bag 18 dollars; the difference is 6 dollars, and 18 dollars remains. Both fractions refer to the original money, not to money remaining after the book purchase.

Compare the relationship as well as the final amount. A correct number alone does not show how you chose the whole.

Check your understanding 4 Use only the relationship you need

Siti gives away 1/3 of the beads in a bag. She then uses 1/4 of the beads remaining to make a bracelet. The bracelet uses 12 beads. How many beads are left in the bag? Is it necessary to find the original count?

Your model or calculations, conclusion and reason

Show answer

Treat the beads present just before the bracelet as four equal parts. One part, used for the bracelet, is 12 beads. Three parts remain, so 3 × 12 = 36 beads are left. The known 12 is a part of this collection, not the whole; 3/4 × 12 = 9 uses the wrong reference whole.

The original count is unnecessary for this question. You may find it to check: there were 48 beads after the first gift, which was two thirds of the original 72 beads. Then 24 were given away, 12 used, and 36 left. Both routes give 36, but the shorter route keeps only the relationship needed for the requested unknown.

Compare your work: did you identify what the 12 labels, use equal parts of that whole, and answer how many beads are left, rather than how many were there originally? A correct final count alone does not establish those choices.

Before you move on

When reading a fraction problem, name the reference whole, the known quantity and the unknown. Decide whether the unknown is a part, the whole, or a difference. Draw only the equal parts needed to show that decision, then check your conclusion in the original story.