Weight, time and money

What this heading covers

Three measures a child meets before they meet a textbook — how heavy, how long it takes, and what it cost — and each is a heading of its own in these syllabuses.

Almost nothing here needs derivation. It needs the unit right, the arithmetic done in the correct base, and a percentage taken on the correct quantity. Each of the three carries one error the paper is built to catch, and those three errors account for most of what is lost in this topic.

Mensuration — perimeter, area, volume and surface area, with the metric units of length — is a heading of its own and has a lesson of its own.

Weight

Unit	Equals
1 kilogram (kg)	1000 grams
1 gram (g)	1000 milligrams
1 quintal	100 kg
1 metric tonne	1000 kg, which is 10 quintals

The quintal and the metric tonne are the two a question is most likely to use, because they are the units a grain merchant actually works in, and they are the two candidates are least sure of.

Mass and weight are not the same thing

Strictly, mass is the quantity of matter in a body and is measured in kilograms; weight is the force gravity exerts on that mass and is measured in newtons. A body carried to the moon keeps its mass and loses most of its weight.

School mathematics and these papers use "weight" for mass throughout, so a question in the mathematics paper that says weight means mass. The distinction belongs to the science paper, where it is asked directly — and knowing which paper you are in is the whole of the trick.

What the questions look like

• Conversion, usually with a decimal: 3.5 kg is 3500 g, and 250 g is 0.25 kg.
• Addition and subtraction in mixed units — 2 kg 750 g plus 1 kg 500 g is 4 kg 250 g, because the grams carried over a thousand, not over a hundred.
• Rate — the cost of 2.5 kg at ₹48 per kg.
• Average weight, where the trap is that an average is recomputed from the total and not by averaging the averages.
• Net, gross and tare — gross weight is the whole thing, tare is the container, and net is the goods: net = gross − tare.

Time

60 seconds to a minute, 60 minutes to an hour, 24 hours to a day, 7 days to a week.

The 24-hour clock

The 24-hour clock runs 00:00 to 23:59. To convert a p.m. time, add 12 — 4:30 p.m. is 16:30. The two that are asked because they are the two that are wrong in most people's heads:

• 12 noon is 12:00.
• 12 midnight is 00:00, and 12:30 a.m. is 00:30.

Duration is a subtraction in base 60

From 9:45 to 11:20 is 1 hour 35 minutes — borrow 60 minutes, not 100. Written out:

  11 h 20 min   →   10 h 80 min
−  9 h 45 min       9 h 45 min
                 ─────────────
                    1 h 35 min

A question that crosses midnight is the harder form: from 10:30 p.m. to 6:15 a.m. is 1 h 30 min to midnight plus 6 h 15 min after it, which is 7 hours 45 minutes.

The calendar

• Days in the months: 31 for January, March, May, July, August, October, December; 30 for April, June, September, November; 28 or 29 for February.
• A leap year has 366 days and falls when the year divides by 4 — except century years, which must divide by 400. So 1900 was not a leap year and 2000 was. That exception is the question.
• Odd days — the remainder after dividing a count of days by 7 — are what a calendar question turns on. An ordinary year has 365 days, so 1 odd day; a leap year has 2. Hence the same date falls one weekday later next year, and two later if a 29 February falls in between.

Period	Odd days
An ordinary year	1
A leap year	2
100 years	5
400 years	0

The last line is worth holding: the calendar repeats exactly every 400 years, which is why a question can be answered without counting through the centuries.

The angle between the hands

A standing favourite, and it has one formula. The minute hand moves 6° a minute; the hour hand moves 30° an hour, which is half a degree a minute. For a time H:M,

angle = | 30H − 5.5M |

and if the result exceeds 180°, subtract it from 360°. At 3:30, that is |90 − 165| = 75° — not 90°, because the hour hand has already moved halfway towards four. Forgetting that the hour hand moves is the error this question exists to catch.

Money

Cost price, selling price, profit, loss, discount and simple interest — and the whole topic turns on which quantity the percentage is taken on.

Profit and loss are reckoned on the cost price

• profit = SP − CP, and loss = CP − SP
• profit % = (profit ÷ CP) × 100, and loss % = (loss ÷ CP) × 100
• SP = CP × (100 + profit %) ÷ 100, and CP = SP × 100 ÷ (100 + profit %)

The last one matters: given a selling price and a profit percentage, you cannot take the percentage off the selling price to get back to cost. ₹500 sold at 25 % profit came from ₹400, not from ₹375.

Discount is reckoned on the marked price

• discount = MP − SP, and discount % = (discount ÷ MP) × 100
• so selling price = marked price − discount, and the cost price does not enter the calculation at all

A question giving a marked price, a discount and a cost price is asking you to do both halves in order: discount off the marked price to reach the selling price, then profit or loss against the cost price.

Two successive discounts do not add. Twenty per cent and then ten per cent is not thirty per cent, because the second is taken on what is left after the first. The single equivalent is

a + b − (ab ÷ 100)

so 20 % and 10 % together come to 28 %, not 30 %.

Simple interest

SI = (P × R × T) ÷ 100          amount = P + SI

The rate is per annum, so a time given in months must be divided by 12 before it is used, and a time given in days by 365. Six months is ½, not 6 — and using 6 multiplies the answer by twelve.

Given	Rearranged
Principal	P = (SI × 100) ÷ (R × T)
Rate	R = (SI × 100) ÷ (P × T)
Time	T = (SI × 100) ÷ (P × R)

A useful consequence to recognise rather than derive: under simple interest a sum doubles when R × T = 100 — so at 5 % it takes 20 years.

The everyday money a primary paper asks about

• Writing amounts in rupees and paise, and 100 paise to the rupee.
• Making a bill: quantity × rate for each line, then the total.
• Giving change, which is a subtraction, and estimating whether the change offered is plausible.
• The unitary method — find the value of one, then multiply. The cost of 7 pens at the rate given for 5 is a unitary-method question, and it is the single most useful technique in the primary paper.

Questions of the kind the paper sets

A shopkeeper buys an article for ₹400 and sells it for ₹500. What is the profit per cent?

Profit = ₹100, on a cost of ₹400. Profit % = (100 ÷ 400) × 100 = 25 %. Dividing by 500 gives 20 %, which will be among the options.

An article is marked at ₹800 and sold at a discount of 15 %. If it cost the shopkeeper ₹600, what is his profit per cent?

Both halves, in order. Discount = 15 % of 800 = ₹120, so SP = ₹680. Profit = 680 − 600 = ₹80 on a cost of ₹600, so profit % = (80 ÷ 600) × 100 = 13⅓ %. Taking the 15 % off ₹600 is the error the question is built on.

Successive discounts of 20 % and 10 % are equivalent to a single discount of —

20 + 10 − (20 × 10 ÷ 100) = 30 − 2 = 28 %.

What is the simple interest on ₹2,000 at 5 % per annum for 6 months?

T = ½ year. SI = (2000 × 5 × ½) ÷ 100 = ₹50. Using 6 instead of ½ gives ₹600.

At what rate per annum will ₹1,200 amount to ₹1,500 in 5 years at simple interest?

SI = 1500 − 1200 = ₹300. R = (300 × 100) ÷ (1200 × 5) = 5 %.

Which of these is not a leap year — 1996, 2000, 1900, 2024?

1900. It divides by 4 but is a century year and does not divide by 400.

15 August 2025 was a Friday. On which day did 15 August 2026 fall?

2026 is not a leap year and no 29 February falls between the two dates, so there is 1 odd day: Saturday.

The angle between the hands of a clock at 3:30 is — (a) 90° (b) 75° (c) 80° (d) 60°

|30 × 3 − 5.5 × 30| = |90 − 165| = 75°.

A train leaves at 22:40 and reaches at 06:15 the next morning. How long is the journey?

1 h 20 min to midnight, then 6 h 15 min — 7 hours 35 minutes.

2 kg 750 g + 1 kg 500 g = (a) 3 kg 250 g (b) 4 kg 250 g (c) 4 kg 150 g (d) 3 kg 1250 g

750 + 500 = 1250 g, which is 1 kg 250 g; carry the kilogram. Option (d) is the same quantity written wrongly, since grams carry at 1000.

8 quintals of rice is — (a) 80 kg (b) 800 kg (c) 8000 kg (d) 0.8 kg

Mistakes that cost marks here

• Calculating profit percentage on the selling price. It is always on the cost price.
• Taking a discount on the cost price. Discount is always on the marked price.
• Adding two successive discounts. 20 % then 10 % is 28 %, not 30 %.
• Using months directly in the simple-interest formula instead of converting to years.
• Treating every century year as a leap year. It must divide by 400.
• Subtracting times in base 10. An hour has 60 minutes, not 100.
• Writing 12 midnight as 12:00 on the 24-hour clock. It is 00:00.
• Forgetting that the hour hand moves when finding the angle between the hands.
• Carrying grams at 100 instead of 1000.
• Confusing the quintal with the metric tonne — 100 kg against 1000 kg.
• Answering a mathematics question with the physics distinction between mass and weight. In this paper "weight" means mass.

In one line

Weight is a chain of thousands, time is arithmetic in base sixty with one calendar exception that divides by 400, and money is a question about which quantity the percentage sits on — profit and loss on the cost price, discount on the marked price, and never the other way round.

For the classroom

All three are best taught with the real thing in the room. A balance and a set of weights makes grams and kilograms a matter of experience rather than of conversion; a classroom shop with price labels, marked prices and a discount sign teaches the difference between marked and selling price in one afternoon, and teaches it in the only order that sticks — discount first, then profit.

For time, give the class a railway timetable and ask how long each journey takes. The base-sixty subtraction stops being a rule to remember the moment a child has worked out that the 22:40 train does not arrive after seven and a half hours of decimal arithmetic.