Cumulative Fractions Practice Test Worksheets
Cumulative fraction questions describe a quantity that changes over several steps. The key is to track each new total, spot the telescoping pattern, and know when to work forwards or backwards.
For students who know fraction basics, but lose marks when several fractions have to be applied one after another.
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What Do Cumulative Fractions Questions Look Like?
Cumulative fraction questions describe a quantity that changes over several steps, growing by a fraction of each new running total. Students must write each growth step as a multiplier, spot where intermediate factors cancel out, and calculate the overall increase or final amount. Here is a worked example.
Use the Telescoping Pattern
Question: A campaign has raised $180 at the end of week 1. Its total grows by a fraction of the previous week's total: 14 in week 2, 15 in week 3, 16 in week 4, and 17 in week 5. How much has the total increased from the end of week 1 to the end of week 5?
Worked Method
Step 1: Convert each growth into a multiplier: 54, 65, 76, 87.
Step 2: Multiply them: 54 × 65 × 76 × 87 = 2.
Step 3: Final total = 180 × 2 = $360, so the increase is 360 - 180 = $180.
Tip: In this pattern, most middle numbers cancel. That is the shortcut.
Rules to Help You Solve Cumulative Fractions Questions
To solve cumulative fraction questions like the campaign example above, write each step's growth as an improper multiplier and look for cancelling terms.
🔁 Rule 1: Add the fraction to the whole
If something grows by 1n, the new total is the old total multiplied by n + 1n.
Example: Grow by 14 → multiply by 54.
🧩 Rule 2: Look for telescoping
Sequences like 54 × 65 × 76 × 87 cancel neatly, leaving a much simpler multiplier.
Example: 54 × 65 × 76 × 87 = 2.
↩️ Rule 3: Reverse with division
If you know the final total, divide by the total multiplier to find the starting value.
Example: If final = 780 and multiplier = 3, start = 260.
🎯 Rule 4: Growth during a step uses the previous total
To find growth during a particular period, first find the total at the end of the previous period.
Example: Month 5 growth of 17 uses the end-of-month-4 total.
The Telescopic Pattern
Some cumulative fraction questions look long because they have several steps, but the multipliers often collapse into one short calculation. That shortcut is called the telescopic pattern.
Why the Pattern Works
When a quantity grows by a fraction of its current value, write the new total as a multiplier. For example, growing by 12 means multiplying by 32, growing by 13 means multiplying by 43, and so on.
Main shortcut: Write out the whole chain before calculating. If the top of one fraction matches the bottom of the next fraction, those middle numbers cancel.
What remains: The starting value, the first denominator, and the last numerator. In the example above, 80 × 32 × 43 × 54 × 65 becomes 80 × 62 = 80 × 3 = 240.
Why it helps: You avoid doing every month or week separately, which saves time and reduces calculation errors.
Two More Worked Examples
Here are two more examples showing how to work backwards from a final total and calculate growth during one specific period.
Work Backwards from the Final Total
Question: A road is 780 m long at the end of week 5. Its length increases by a fraction of the previous week's length: 12 in week 2, 13 in week 3, 14 in week 4, and 15 in week 5. What was its length at the end of week 1?
Worked Method
Step 1: Add each growth fraction to 1 to find the multipliers: 32, 43, 54, and 65.
Step 2: Combine the multipliers: 32 × 43 × 54 × 65 = 62 = 3.
Step 3: Work backwards by dividing the final length by 3: 780 ÷ 3 = 260 m.
Tip: Backwards questions undo the growth multiplier. Do not multiply again.
Growth During One Period
Question: A vine is 160 cm tall at the end of month 1. It grows by a fraction of its height at the end of the previous month: 14 in month 2, 15 in month 3, 16 in month 4, and 17 in month 5. How much does it grow during month 5?
Worked Method
Step 1: First find the total at the end of month 4: 160 × 54 × 65 × 76 = 160 × 74 = 280 cm.
Step 2: Month 5 growth is 17 of the month 4 total: 280 × 17 = 40 cm.
The trap: Do not take 17 of the starting value. The growth is based on the previous total.
