Cumulative Fractions Practice
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.
The Rules Every Cumulative Fractions Question Uses
Each step grows by a fraction of the previous total, not the starting amount. That makes the new total a multiplier: growing by one quarter means multiplying by 54.
🔁 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.
How to Solve Cumulative Fractions Questions
Three useful moves: build the multiplier, work backwards from a final total, and isolate growth during one period.
1. Use the Telescoping Pattern
Question: A campaign is $180 at the end of week 1. It grows by 14, then 15, then 16, then 17. How much has it increased by 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.
2. Work Backwards from the Final Total
Question: A road is 780 m at the end of week 5. The total multiplier from week 1 to week 5 is 3. What was its length at the end of week 1?
Worked Method
Step 1: The question tells you the end value and the multiplier from start to finish.
Step 2: Work backwards by dividing: 780 ÷ 3 = 260.
Tip: Backwards questions undo the growth multiplier. Do not multiply again.
3. Growth During One Period
Question: A vine is 160 cm at the end of month 1. It follows the 14, 15, 16, 17 pattern. 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.
Before You Buy
Want to check the level and layout first? Download the free 3-question sample. It uses the same question style, printable format, and answer-key approach as the full pack.
Download Free Sample PDFGet the Full Practice Pack
The full Cumulative Fractions pack contains 90 questions across 3 printable test sets. Students practise forward growth, backwards calculations, comparisons, and growth during a single step.
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