Unit 2 · Topic 02 · Comparing Quantities
Zara's plant grew from 40 cm to 50 cm. She said it grew by 25%. Kabir said it grew by 20%, since 10 out of 50 is 20%.
Both had divided 10 by a number. They just divided by different numbers.
Only one of them had picked the right one.
Zara's tomato plant grew from 40 cm to 50 cm over a month, and she wanted to report the percentage increase for her science journal. She calculated the change first: 50 − 40 = 10 cm. Then she divided the change by the ORIGINAL height: 10/40 × 100 = 25%.
Kabir, checking her work, divided 10 by the NEW height instead: 10/50 × 100 = 20%. "I get a different answer," he said. Zara explained that percentage change is always measured against the original (starting) value, not the new one — because the question is 'how much did it grow relative to where it started,' not relative to where it ended up.
They tested the reverse case too: if the plant had SHRUNK from 50 cm to 40 cm, the decrease would be 10/50 × 100 = 20% — a different percentage from the 25% increase in the other direction, even though the raw change (10 cm) was identical both times. "So growing back up wouldn't undo the percentage drop," Kabir realised. "20% down from 50 gets you to 40, but 25% up from 40 is needed to get back to 50 — they're not symmetric."
Zara then measured her plant with a slightly worn ruler and got 49.5 cm instead of the true 50 cm. The percentage error was |49.5 − 50|/50 × 100 = 1%. "Error works exactly like percentage change," she said, "except you always divide by the TRUE or actual value, and you take the difference as a positive amount regardless of direction."
Kabir tried a trickier one: a recipe called for 200g of flour, but he measured 210g. Error = |210−200|/200 × 100 = 5%. "Same formula every time," he said. "Difference over the original or true value, times 100, and I keep the sign only if the question actually asks whether it's an increase, decrease, or an error size."
By the end, both had one formula they trusted for every version of this problem: change divided by the starting or true value, times 100 — never divided by the ending or measured value instead.
Percentage increase = (new value − original value)/original value × 100. Percentage decrease = (original value − new value)/original value × 100. Both always divide by the ORIGINAL value.
A percentage increase followed by the same-size percentage decrease does not return you to the start, and vice versa, because each percentage is calculated against a different base value.
Percentage error = |measured value − true value|/true value × 100 — always divide by the TRUE (actual/expected) value, and the numerator is a positive difference (an absolute value).
The same core formula — difference over the reference value, times 100 — underlies percentage increase, decrease, and error; the only thing that changes is which value is the 'reference' (original for change, true value for error).
A price rises from ₹80 to ₹100, then later falls from ₹100 back toward ₹80. Compute the percent rise and the percent fall needed to return, and confirm they are different numbers.
You measure a table as 122 cm when its true length is 120 cm. Compute the percentage error.
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