Search
Subjects and activities

Class 12 · Mathematics and money

Check a result a second way

Your calculation says each student uses 3,000 litres of water a day at school. Would you believe it?

What you will learn: Validate a result with an estimate, a second method or units.

What should I know first?

If this feels new, try these first. It is fine to go back.

Small test at the end: scroll down to the end of this page for five short questions that check what you have learnt. Tap an answer, then press Check answer. Go to the questions ↓

Jump to: Go deeper · Check yourself · For grown-ups · For teachers

Safety first

Paper and calculator only.

Keep it safe:

  • Paper and calculation only. Read meters only with permission and never open meter boxes or pipes.

You will need

  • Paper and a calculator
  • A result from your own workOr the made-up figures on this page.

Time: About 40 minutes.

What do you think?

A calculation says each student uses 3,000 litres of water a day at school. Do you believe it? What is your reason?

Make a prediction and write one reason for it. Then tap the guess closest to yours to see how to test it. It is fine to change your mind after checking.

Yes, the calculator worked it out

Write your reason. Then picture it: a bucket holds about 10 litres. How many buckets would that be?

No, it seems far too big

Too big compared with what? Write the comparison, then find a second way to check the figure.

It seems a bit small

Write your reason. Then estimate a student's day: drinking, washing hands, toilet flushes. Add it up and compare.

Not sure

That is fine. Estimate first: about how many buckets of water might one student use in a school day?

Need a hint?

Compare with something you know. A bucket holds about 10 litres; a toilet flush uses roughly 6 to 9 litres.

The story

Tara's spreadsheet says each student uses 3,000 litres of water a day at school. Tanmay whistles. "That's 300 buckets each!" Dhruv frowns. "Nobody uses 300 buckets of water at school." Is the number wrong, and how could Tara check it?

The idea

Before you trust a result, check it a second way. A calculator does exactly what you type, including mistakes.

An order-of-magnitude estimate asks: should the answer be about 1, 10, 100 or 1,000? A bucket holds about 10 litres, so a student using 300 buckets a day is not reasonable.

A second method reaches the same quantity by a different route. A units check asks whether the answer comes out in the units the question asked for. For a speed, metres per second become km/h by multiplying by 3.6, because 1 m/s is 3.6 km in an hour.

When two independent checks agree, the result is more likely to be right, though still not certain. When they disagree, find the mistake before you report anything.

A diagram of two checks on one result. The result says 3,000 litres per student per day. Check 1, an estimate: a bucket is about 10 litres, so 10 to 50 litres is reasonable, not 3,000. Check 2, a different method: 3,000 litres divided by 100 students is 30 litres each. Verdict: the total had not been divided by the number of students; corrected to 30 litres.Two checks on one resultResult: 3,000 Lper student per dayCheck 1: estimatea bucket is about 10 L,so 10–50 L, not 3,000Check 2: method3,000 L ÷ 100 students= 30 L eachVerdict: the total was notdivided by the number ofstudents. Corrected: 30 L.
A diagram of two checks on one result. The result says 3,000 litres per student per day. Check 1, an estimate: a bucket is about 10 litres, so 10 to 50 litres is reasonable, not 3,000. Check 2, a different method: 3,000 litres divided by 100 students is 30 litres each. Verdict: the total had not been divided by the number of students; corrected to 30 litres.

Worked example

Fixing the water figure (made-up school data)

  1. Claim: 3,000 L per student per day.
  2. Estimate: about 10–50 L seems reasonable. 3,000 L is about 100 times too big.
  3. Second method: the school meter showed 3,000 L for the day, shared by 100 students: 3,000 ÷ 100 = 30 L each.
  4. Units: litres per student per day, as the question asked.
  5. Meaning: 30 L is an average share, not what each student personally used. The same meter also counts cleaning, staff, the canteen and any leaks.

The first figure was the school total, not the amount per student. The checked result is 30 L per student per day on average: a fair share of the total, not a measurement of any one student.

Try it

  1. Write the result.

    Copy the number, its units and what it is supposed to mean.

  2. Estimate.

    Is the answer about 1, 10, 100 or 1,000? Use a simple fact you know, such as a bucket of about 10 L.

  3. Use a second method.

    Reach the same quantity a different way, or work backwards from the answer.

  4. Check the units.

    Do the units match what the question asked?

  5. Give a verdict.

    Agree, or find and correct the mistake. Write what you changed.

What you make (paper or offline)

Your validation sheet

A diagram of a result checked by an estimate and a second method, with a verdict.Two checks on one resultResult: 3,000 Lper student per dayCheck 1: estimatea bucket is about 10 L,so 10–50 L, not 3,000Check 2: method3,000 L ÷ 100 students= 30 L eachVerdict: the total was notdivided by the number ofstudents. Corrected: 30 L.
A diagram of a result checked by an estimate and a second method, with a verdict.

The original result with units.

Two checks, worked in full.

Your verdict and any correction.

Make it on paper. Keep it with you. Nothing to upload.

What did we find out?

Read this after you have tried it. Look back at your guess and your reason first.

3,000 L is about 300 buckets each: far too big. The 3,000 L was the whole school's meter reading. Shared by 100 students, it is 30 L each.

Which check would have caught the mistake fastest: an estimate, a second method, or the units?

Go deeper

Plausible is not the same as validated

A plausible result passes a rough estimate: it is the right size. A validated result also agrees with an independent check, one that does not use the same data or the same assumptions.

Two methods that start from the same faulty reading will agree and both be wrong. Independence matters more than the number of checks.

Worked example: is 30 L per student validated?
  1. Result: the meter showed 3,000 L for the day; 3,000 ÷ 100 students = 30 L each. It passes the estimate of 10 to 50 L, so it is plausible.
  2. Assumption: all the water was used by students. But the same meter also supplies the canteen and the garden.
  3. Independent check: in one block, a class counted tap and toilet use for one hour. 40 students used about 160 L: 4 L each an hour, so about 24 L over a 6-hour day.
  4. The two figures are close. The gap of about 6 L each (600 L for the school) fits canteen and garden use.

About 24 to 30 L per student: now supported by a check that does not share the meter's assumption.

Challenge 1

Two classmates work out water per student from the same meter reading, one by hand and one with a spreadsheet. They agree. Is the result validated?

Work it out on paper and write your reason before you look.

See a worked answer
  1. No. Both used the same reading and the same assumption (all water is used by students).
  2. They checked only the arithmetic.
  3. An independent check uses different data, such as a tap count or the drop in the water tank level.
Challenge 2

A report says: "Our school wastes 1,000 L a day from dripping taps." It assumed 50 taps each dripping 20 L a day. Which assumption would you test first, and how?

Work it out on paper and write your reason before you look.

See a worked answer
  1. The answer depends most on how many taps really drip. Walk round and count them.
  2. Then measure one drip with a cup for 10 minutes, as in the Class 4 dripping tap, and scale up.
  3. If only 5 taps drip, the waste is about 100 L a day: still worth fixing, but a tenth of the claim.

Two of the questions in "Check yourself" use this part.

Check yourself

Five questions about what this page teaches. Choose an answer and press Check answer, or open "See the answer and why". Nothing is saved or sent: your choices stay on this screen and disappear when you leave or reload the page.

  1. Question 1 of 5The school used 3,000 L in a day, shared by 100 students. How much is that per student on average?

    See the answer and why
    • A. 0.3 L Not this one. That divides by 10,000, not by 100.
    • B. 3 L Not this one. That divides by 1,000, not by 100.
    • C. 30 L Correct. 3,000 ÷ 100 = 30 L each on average. It is a share of the total, not what each student actually used.
    • D. 300 L Not this one. That divides by 10, not by 100.
  2. Question 2 of 5A runner covers 100 m in 20 s. What is that speed in km/h?

    See the answer and why
    • A. 20 km/h Not this one. 20 is the time in seconds, not a speed.
    • B. 18 km/h Correct. 100 ÷ 20 = 5 m/s, and 5 × 3.6 = 18 km/h.
    • C. 5 km/h Not this one. 5 is the speed in metres per second, not km/h.
    • D. 36 km/h Not this one. That doubles the correct answer: 5 × 3.6 is 18.
  3. Question 3 of 5Each tile covers 0.25 m², so 4 tiles cover 1 m². Someone says 40 tiles cover 160 m². What is the correct area?

    See the answer and why
    • A. 40 m² Not this one. That counts tiles, not their area.
    • B. 10 m² Correct. 4 tiles make 1 m², so 40 tiles make 40 ÷ 4 = 10 m². Check: 40 × 0.25 = 10.
    • C. 160 m² Not this one. That multiplies by 4 instead of dividing: the estimate shows it is far too big.
    • D. 4 m² Not this one. That uses 0.1 m² per tile instead of 0.25 m².
  4. Question 4 of 5Two students work out water per student from the same meter reading, one by hand and one with a spreadsheet. They agree. What has been checked?

    See the answer and why
    • A. The result is now fully validated, because two methods agree Not this one. The methods are not independent: they share the same data.
    • B. Only the arithmetic Correct. Both used the same reading and the same assumptions.
    • C. That the meter is accurate Not this one. Nobody tested the meter.
    • D. That students use the same water every day Not this one. One day's reading cannot show that.
  5. Question 5 of 5A report says 50 taps each waste 20 L a day: 1,000 L. Which check should come first?

    See the answer and why
    • A. Count how many taps really drip Correct. The total depends most on the number of dripping taps.
    • B. Redo 50 × 20 on a calculator Not this one. The arithmetic is already right. The assumption is the weak part.
    • C. Write the answer as 1 kL instead Not this one. Changing the unit does not test the figure.
    • D. Ask the principal whether 1,000 L sounds right Not this one. An opinion is not a check.

Think back

Which check found the mistake fastest: the estimate or the second method?

What does the 30 L average tell us, and what can it not tell us?

Where in your own work could a total have been mistaken for an amount per person?

Investigate further

  • Validate a figure from a news story with a rough estimate.
  • Write one second method for a calculation from another subject.

Can you use it here?

A website says charging one phone uses 500 kWh a year. A charger draws about 10 W for about 2 hours a day. Check the claim.

Ask someone at home

Ask someone at home to guess how many litres a bucket bath uses. Then measure it with a 1 litre bottle.

No one free to join in? That's fine. You can do it by yourself with drawings, and ask a grown-up to help with the materials.

For grown-upsTap a heading to open it. The safety notes are at the top of the page.

Before you start
  • A calculation from homework or a project to check.
Questions that help
  • "Is that answer about the size you would expect?"
  • "Can you get the same number another way?"
What's really going on

Engineers, accountants and scientists all check results a second way before they rely on them.

Other ways to do it
  • Check a shopping bill by estimating it first.

For teachersLearning aims, classroom ideas and what to look for.

Learning aims
  • Validate a result using an estimate, a second method and a units check.
  • Decide what agreement between checks does and does not show.
Subject links
  • Mathematics: estimation and units.
  • Science: data checking.

These links are general. They have not been checked against a specific board syllabus.

In the classroom
  • Swap results with a partner and validate each other's.
What to look for
  • Two independent checks shown.
  • A clear verdict and correction.
ProLearners Assist