Mental Ability · Lesson 5

Decimals & Money

Learn what the little dot in 4.75 really means, how to compare and add decimals without falling into traps, and how to handle rupees, paise, bills, profit and "cost of one" problems like a pro.

~40 min read8 quick checksTest: 20 questions

1Tenths and hundredths

A decimal is a way of writing parts of a whole using a dot called the decimal point. Everything to the left of the point is whole ones; everything to the right is a part smaller than one.

Cut one whole into 10 equal parts. Each part is one tenth = 1⁄10 = 0.1. Three such parts make 3⁄10 = 0.3 (read "zero point three").

3 out of 10 parts = 3/10 = 0.3
One whole cut into 10 equal parts: each part is 0.1 (one tenth).

Now cut the whole into 100 equal parts. Each tiny part is one hundredth = 1⁄100 = 0.01. A hundred-square grid shows this beautifully:

0.5 = 50 squares
0.5 = 5 tenths = 50 hundredths
0.45 = 45 squares
0.45 = 4 tenths + 5 hundredths

Each column of the grid is one tenth (10 squares). So 0.5 fills 5 whole columns, while 0.45 fills only 4 columns and half of the next. Look: 0.5 is more, even though "45" looks bigger than "5". Keep this picture in mind — it beats a very common trap later.

In wordsFractionDecimal
seven tenths7/100.7
seven hundredths7/1000.07
twenty-three hundredths23/1000.23
eighty-nine hundredths89/1000.89
four and six tenths4 6/104.6
twelve and five hundredths12 5/10012.05
📌 Remember

10 tenths = 1 whole and 10 hundredths = 1 tenth. So 100 hundredths = 1 whole. A zero in front, like 0.6, simply says "no whole ones".

What does the 7 in 0.07 stand for?

  1. 7 tenths
  2. 7 hundredths
  3. 7 ones
  4. 70 hundredths
The 7 is in the second place after the point — the hundredths place. 0.07 = 7/100.

2Decimal place value

The place value chart simply keeps going past the ones place. Every place to the right is 10 times smaller than the place before it.

TensOnes•TenthsHundredthsThousandths 52.738 5020.70.030.008
52.738 = 50 + 2 + 0.7 + 0.03 + 0.008 (the bottom row shows each digit's place value)
✏️ Worked example — build a decimal

Write as a decimal: 6 tens, 3 ones, 9 hundredths.

  1. 6 tens = 60, 3 ones = 3 → whole part is 63.
  2. There are no tenths, so the tenths place gets a 0.
  3. 9 hundredths goes in the second place: 63.09.
  4. Answer: 63.09 (not 63.9 — that would be 9 tenths!).

Multiplying and dividing by 10, 100, 1000

Because each place is 10 times the next, multiplying by 10 moves every digit one place to the left — which looks like the decimal point hopping one place to the right.

OperationPoint movesExample
× 101 place right4.8 × 10 = 48
× 1002 places right0.36 × 100 = 36 ; 5.2 × 100 = 520
× 10003 places right2.625 × 1000 = 2625 ; 0.4 × 1000 = 400
÷ 101 place left57.4 ÷ 10 = 5.74
÷ 1002 places left245 ÷ 100 = 2.45 ; 8 ÷ 100 = 0.08
÷ 10003 places left3500 ÷ 1000 = 3.5
Remember: count the zeros! × 100 has two zeros, so the point hops two places. Multiply → hop right (number grows). Divide → hop left (number shrinks). If you run out of digits, fill the gap with zeros: 5.2 × 100 = 520.
✏️ Worked example — olympiad style

Which is equal to 7.5 × 100: (a) 75 ÷ 10 (b) 0.75 × 1000 (c) 7500 ÷ 100 (d) 750 ÷ 100?

  1. First find the target: 7.5 × 100 → point hops 2 places right → 750.
  2. (a) 75 ÷ 10 = 7.5 ✗
  3. (b) 0.75 × 1000 → hop 3 right → 750 ✓
  4. (c) 7500 ÷ 100 = 75 ✗ ; (d) 750 ÷ 100 = 7.5 ✗
  5. Answer: (b). Work out the target first, then test every option.

What is the place value of 4 in 23.548?

  1. 4
  2. 0.4
  3. 0.04
  4. 0.004
After the point: 5 is in tenths, 4 is in hundredths, 8 is in thousandths. So 4 × 0.01 = 0.04.

3Fractions ↔ decimals

A decimal is really a fraction with denominator 10, 100 or 1000. So the trick for turning a fraction into a decimal is: make the denominator 10 or 100.

✏️ Worked example — fraction to decimal

Write 3/4 and 7/20 as decimals.

  1. 3/4: what times 4 gives 100? 4 × 25 = 100. Multiply top and bottom by 25 → 75/100 = 0.75.
  2. 7/20: 20 × 5 = 100. Multiply top and bottom by 5 → 35/100 = 0.35.
  3. Mixed numbers: 2 3/5 → 3/5 = 6/10 = 0.6, so 2 3/5 = 2.6.
✏️ Worked example — decimal to fraction
  1. 0.6 = 6/10 → divide top and bottom by 2 → 3/5.
  2. 0.25 = 25/100 → divide by 25 → 1/4.
  3. 0.08 = 8/100 → divide by 4 → 2/25.
  4. 1.5 = 1 5/10 = 1 1/2.

Learn these by heart — they appear again and again:

FractionDecimalFractionDecimal
1/20.51/100.1
1/40.251/200.05
3/40.751/250.04
1/50.21/500.02
2/50.41/1000.01
3/50.64/50.8
00.10.20.30.40.50.60.70.80.91 0.09 0.45 0.5 0.8
On a number line, the number further to the right is bigger: 0.09 < 0.45 < 0.5 < 0.8.

What is 2/5 written as a decimal?

  1. 0.25
  2. 2.5
  3. 0.4
  4. 0.04
2/5 = 4/10 (multiply top and bottom by 2) = 0.4. Don't just write the two digits next to each other!

4Comparing and ordering decimals

To compare decimals, line up the decimal points and compare place by place from the left — just like comparing whole numbers.

✏️ Worked example — the equal-digits trick

Arrange 0.6, 0.06, 0.66 and 0.606 in ascending order (smallest first).

  1. Give them all the same number of decimal places by adding zeros at the end (this doesn't change the value): 0.600, 0.060, 0.660, 0.606.
  2. Now forget the point and compare them like whole numbers: 600, 60, 660, 606.
  3. Order: 60 < 600 < 606 < 660.
  4. Answer: 0.06, 0.6, 0.606, 0.66.
⚠️ Olympiad trap — "longer means bigger"

Many students think 0.45 > 0.5 because 45 > 5, or that 0.09 > 0.5 because 9 > 5. Both are wrong! Compare the tenths first: 0.5 has 5 tenths; 0.45 has only 4 tenths; 0.09 has 0 tenths. So 0.5 is the greatest. Write 0.5 as 0.50 and the trap disappears: 50 hundredths > 45 hundredths.

⚠️ Olympiad trap — zeros that matter and zeros that don't

Zeros at the end after the point change nothing: 0.5 = 0.50 = 0.500. But a zero right after the point changes everything: 0.5 ≠ 0.05. And 3.07 ≠ 3.7.

✏️ Worked example — mixing fractions and decimals

Which is smallest: 3/4, 0.7, 1/2 + 0.3, 16/20?

  1. Turn everything into decimals: 3/4 = 0.75.
  2. 0.7 stays 0.7 (= 0.70).
  3. 1/2 + 0.3 = 0.5 + 0.3 = 0.8.
  4. 16/20 = 80/100 = 0.8.
  5. Compare 0.75, 0.70, 0.80, 0.80 → smallest is 0.7.

Which of these is the smallest?

  1. 0.3
  2. 0.29
  3. 0.303
  4. 0.31
Write all with 3 places: 0.300, 0.290, 0.303, 0.310. The smallest is 0.290 = 0.29.

5Adding and subtracting decimals

The one golden rule: line up the decimal points one under the other. Then ones sit under ones, tenths under tenths, hundredths under hundredths. Fill empty places with zeros, add or subtract as usual, and bring the point straight down.

✗ Wrong 5.4 − 0.89 points not in one column ✓ Right 5.40 − 0.89 4.51 add a 0, points in one column
5 ones 4 tenths − 89 hundredths = 5.40 − 0.89 = 4.51
✏️ Worked example — adding three decimals

Add 14.6 + 2.35 + 0.9.

  1. Make equal places: 14.60 + 2.35 + 0.90.
  2. Hundredths: 0 + 5 + 0 = 5.
  3. Tenths: 6 + 3 + 9 = 18 → write 8, carry 1 to the ones.
  4. Ones: 4 + 2 + 0 + 1 (carry) = 7. Tens: 1.
  5. Answer: 17.85.
✏️ Worked example — subtracting from a whole number

Find 9 − 2.46.

  1. A whole number has a hidden point at its end: 9 = 9.00.
  2. 9.00 − 2.46: hundredths 0 − 6 can't be done, so borrow along: 9.00 becomes 8 ones, 9 tenths, 10 hundredths.
  3. 10 − 6 = 4 ; 9 − 4 = 5 ; 8 − 2 = 6.
  4. Answer: 6.54. Check: 6.54 + 2.46 = 9.00 ✓
💡 Shortcut — check by adding back

After subtracting, add your answer to the smaller number. You must get the bigger number back. It takes five seconds and catches most borrowing mistakes.

Find 8 − 3.27.

  1. 5.83
  2. 4.73
  3. 5.27
  4. 4.83
8.00 − 3.27 = 4.73. Check: 4.73 + 3.27 = 8.00 ✓

6Money: rupees and paise

Indian money uses rupees (₹) and paise (p). ₹1 = 100 paise — so paise are exactly hundredths of a rupee, and money is written with two digits after the point: ₹25.50 means 25 rupees and 50 paise.

₹ → paise

Multiply by 100 (point hops 2 places right).
₹8.75 = 875 p
₹6.05 = 605 p
₹12 = 1,200 p

paise → ₹

Divide by 100 (point hops 2 places left).
350 p = ₹3.50
1,625 p = ₹16.25
75 p = ₹0.75

⚠️ Olympiad trap — ₹5.5 is not "5 rupees 5 paise"

₹5.5 = ₹5.50 = 5 rupees 50 paise = 550 p. But 5 rupees 5 paise = ₹5.05 = 505 p. Paise always fill two places, so single-digit paise need a 0 in front.

✏️ Worked example — which coins are worth most? (real olympiad question)

(A) 6 fifty-paise coins (B) 4 one-rupee coins (C) 3 two-rupee coins (D) 20 twenty-five-paise coins

  1. (A) 6 × 50 p = 300 p = ₹3.
  2. (B) 4 × ₹1 = ₹4.
  3. (C) 3 × ₹2 = ₹6.
  4. (D) 20 × 25 p = 500 p = ₹5.
  5. Greatest: (C). Tip: 2 fifty-paise coins make ₹1, and 4 twenty-five-paise coins make ₹1.
🌟 Did you know?

Coins smaller than 50 paise (like 25 paise) stopped being used in India in 2011. Olympiad questions sometimes still use them — just do the maths.

1,005 paise is equal to:

  1. ₹100.5
  2. ₹10.50
  3. ₹10.05
  4. ₹1.005
Divide by 100: 1005 ÷ 100 = 10.05. So it is 10 rupees and 5 paise = ₹10.05.

7Bills and change

A bill lists each item, its quantity, its rate (price of one) and the amount. Amount = quantity × rate. The total is the sum of all amounts.

ItemQuantityRateAmount
Notebook2₹35.50₹71.00
Box of crayons1₹48.75₹48.75
Pencil4₹6.25₹25.00
Total₹144.75
✏️ Worked example — finding the change

Riya pays the bill above with a ₹200 note. How much change does she get?

  1. Check the total: 71.00 + 48.75 + 25.00 = 144.75.
  2. Change = money given − total = 200.00 − 144.75.
  3. Count up (shopkeeper's method): 144.75 → 145.00 is 25 p; 145 → 200 is ₹55.
  4. Change = ₹55 + 25 p = ₹55.25.
✏️ Worked example — tickets for a family

At a museum, an adult ticket costs ₹45.50 and a child ticket costs ₹30.25. What do 2 adults and 3 children pay?

  1. Adults: 2 × 45.50 = ₹91.00.
  2. Children: 3 × 30.25 = ₹90.75.
  3. Total: 91.00 + 90.75 = ₹181.75.
  4. Trap: read carefully who is an adult! "Mr and Mrs Rao and their 3 children" means 2 adults + 3 children.
💡 Shortcut — multiply money by splitting

3 × ₹30.25 = 3 × ₹30 + 3 × 25 p = ₹90 + 75 p = ₹90.75. Rupees and paise separately is often quicker in your head.

8Profit and loss

A shopkeeper buys things and then sells them.

Cost price (CP)
The price at which the shopkeeper buys an item.
Selling price (SP)
The price at which the shopkeeper sells it.
Profit (gain)
When SP is more than CP. Profit = SP − CP.
Loss
When SP is less than CP. Loss = CP − SP.
Remember: always subtract smaller from bigger. If Selling is bigger → Smile (profit). If cost is bigger → sad (loss).
✏️ Worked example — profit

A shopkeeper buys a cycle for ₹2,450 and sells it for ₹2,700. Profit or loss, and how much?

  1. SP (₹2,700) > CP (₹2,450) → profit.
  2. Profit = 2,700 − 2,450 = ₹250.
✏️ Worked example — loss on many items

Raju buys 12 mangoes at ₹15 each and sells all of them together for ₹162. Find his profit or loss.

  1. Total CP = 12 × 15 = ₹180.
  2. SP = ₹162, which is less than ₹180 → loss.
  3. Loss = 180 − 162 = ₹18.
✏️ Worked example — olympiad twist

A shopkeeper buys 10 pens for ₹85 and sells each pen for ₹9.50. What is his profit?

  1. Compare the same quantity: SP of 10 pens = 10 × 9.50 = ₹95.
  2. CP of 10 pens = ₹85.
  3. Profit = 95 − 85 = ₹10. (Trap: subtracting 9.50 from 85 mixes 1 pen with 10 pens!)

A grocer buys 20 kg of sugar at ₹42 per kg and sells it at ₹45 per kg. What is his total profit?

  1. ₹900
  2. ₹3
  3. ₹840
  4. ₹60
CP = 20 × 42 = ₹840, SP = 20 × 45 = ₹900. Profit = 900 − 840 = ₹60. (Quicker: ₹3 profit per kg × 20 kg = ₹60. ₹3 is the profit on just one kg.)

9Unitary method and shop offers

The unitary method means: first find the value of ONE, then multiply to get the value of as many as you need.

6 notebooks₹180 1 notebook₹30 4 notebooks₹120 ÷ 6× 4 Divide to find ONE, then multiply for MANY
The unitary method in two hops.
✏️ Worked example — with decimals

8 m of cloth costs ₹340. What do 5 m cost?

  1. Cost of 1 m = 340 ÷ 8 = ₹42.50 (8 × 42 = 336, remaining ₹4 = 400 p, 400 ÷ 8 = 50 p).
  2. Cost of 5 m = 5 × 42.50 = ₹212.50.
✏️ Worked example — rate problems (real olympiad question)

On a school trip there are 2 teachers for every 9 pupils. How many teachers are there for 216 pupils?

  1. How many groups of 9 pupils? 216 ÷ 9 = 24 groups.
  2. Each group has 2 teachers → 24 × 2 = 48 teachers.
✏️ Worked example — "Buy 2, get 1 free"

A packet of biscuits costs ₹10. The offer is Buy 2, get 1 free. How many packets can you take home with ₹60?

  1. Packets you pay for: 60 ÷ 10 = 6.
  2. Every 2 paid packets give 1 free: 6 ÷ 2 = 3 free packets.
  3. Total = 6 + 3 = 9 packets.
⚠️ Olympiad trap — offers

"Buy 2, get 1 free" means you get 3 packets for the price of 2 — not 2 packets for the price of 1. Always count paid packets first, then add the free ones. Students who stop at "6 packets" lose the mark.

✏️ Worked example — comparing (swap trick, real olympiad question)

5 teacups cost as much as 2 teapots. 4 teacups and 2 teapots cost ₹198. Find the cost of 1 teapot.

  1. Swap the 2 teapots for 5 teacups (same cost): 4 cups + 5 cups = 9 teacups cost ₹198.
  2. 1 teacup = 198 ÷ 9 = ₹22.
  3. 2 teapots = 5 teacups = 5 × 22 = ₹110, so 1 teapot = ₹55.
  4. Check: 4 × 22 + 2 × 55 = 88 + 110 = 198 ✓

A pen costs ₹20. The offer is "Buy 4, get 1 free". How many pens can Kabir get for ₹160?

  1. 10
  2. 8
  3. 9
  4. 12
Paid pens = 160 ÷ 20 = 8. Every 4 paid give 1 free → 8 ÷ 4 = 2 free. Total = 8 + 2 = 10.

Watch and explore

10Quick recap

Flip cards

QuestionWhich is greater: 0.5 or 0.45?
Answer0.5 (= 0.50). 50 hundredths > 45 hundredths.
Question3.049 × 100 = ?
Answer304.9 (point hops 2 places right).
Question7/20 as a decimal?
Answer7/20 = 35/100 = 0.35
Question6.3 − 2.58 = ?
Answer6.30 − 2.58 = 3.72
Question4,208 paise in rupees?
Answer₹42.08
QuestionCP ₹560, SP ₹495. Profit or loss?
AnswerLoss of ₹65 (560 − 495).
Question4 kg of apples cost ₹250. Cost of 1 kg?
Answer250 ÷ 4 = ₹62.50
Question8 fifty-paise coins = ?
Answer400 p = ₹4