Unit 5 · Topic 02 · Decimals
A tailor's price list showed cloth at Rs 3/4 per metre in one column and Rs 0.75 per metre in another.
A customer complained the shop was showing two different prices for the same cloth.
The shopkeeper's daughter, Meher, had to prove — right there at the counter — that 3/4 and 0.75 were exactly the same amount of money.
Meher helped out at her mother's cloth shop on weekends, and one afternoon a customer pointed at the price board, annoyed. One list, handwritten, said 'silk trim: Rs 3/4 per metre.' The till receipt printer, which only understood decimals, printed 'Rs 0.75 per metre.'
"Which is it?" the customer asked. "Are you charging me two different prices?"
Meher's mother was busy with another customer, so Meher had to answer. She knew both numbers were meant to be the same — her mother always said so — but she had never actually proven it.
She grabbed a scrap of paper. "A fraction is just a division that hasn't been done yet," her mother had told her once. So she divided: 3 ÷ 4.
3 does not go into 4, so she added a decimal point and a zero: 30 ÷ 4 = 7 remainder 2. Write down 7. Bring down another zero: 20 ÷ 4 = 5 exactly.
0.75. She showed the customer both numbers side by side and the working underneath.
"Same price," she said. "Different-looking labels on the same amount, like a length in centimetres and the same length in metres."
The customer, satisfied, then asked something that stumped Meher for longer: "What about a third of a metre? What's that as a decimal?"
She tried the same division: 1 ÷ 3. 10 ÷ 3 = 3 remainder 1. Bring down a zero: 10 ÷ 3 = 3 remainder 1 again. And again.
It never stopped. 0.333... forever.
That evening she asked her mother about it. "Some fractions land exactly, like quarters. Others go on forever, like thirds. It depends on whether the bottom number can be built entirely from 2s and 5s. Four is 2 times 2 — lands exactly. Three isn't — it repeats."
Meher wrote both kinds into her notebook that night: 'terminating' for the ones that stop, 'recurring' for the ones that go on. She underlined the shop's silk trim price twice — the first fraction she had ever proven true with her own hand.
A fraction is a division waiting to happen. To turn any fraction into a decimal, divide the numerator by the denominator using long division, adding zeros after the decimal point as needed.
Some conversions land exactly and STOP — these are called terminating decimals. This happens whenever the denominator's only prime factors are 2 and 5, because tenths, hundredths and so on are all built from 2s and 5s. 1/4 = 0.25 stops because 4 = 2×2.
Other conversions never stop and instead repeat a digit or block of digits forever — these are recurring decimals, like 1/3 = 0.333... Both are correct; they simply describe different kinds of fractions.
To go the other way, count the decimal places to choose the denominator: one place means tenths, two places means hundredths, three means thousandths — then simplify the resulting fraction to its lowest terms.
Convert 4/10, 67/10, and 241/100 to decimal form.
Make 3/5 and 9/20 into decimal fractions, then write decimals.
Write 0.6, 0.37, and 5.4 as fractions using place-count rule.
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