Class 7 Mathematics Chapter 12: Another Peek Beyond the Point

A strip three tenths of a metre wide

A floor strip is 0.4 metres long and 0.3 metres wide. What area does it cover?

Area is still length times width, so the sum is 0.3 times 0.4. The question is what that comes to.

Start with the whole numbers

Here is the method. Ignore the points for a moment and multiply the digits as whole numbers.

Three times 4 is 12. That is the right string of digits, and all that remains is to decide where the point goes.

Multiplying by a tenth

To see where, think about what 0.3 means. It is 3 tenths, so multiplying by it is multiplying by 3 and then dividing by 10.

In the same way, multiplying by 0.4 is multiplying by 4 and dividing by 10 again.

So the point moves twice

Altogether we multiplied by 3 and by 4, which gave the 12, and then divided by 10 twice over.

Dividing by 10 twice moves the point two places to the left, so 12 becomes 0.12. The strip covers 0.12 square metres.

Counting the decimal places

That gives a rule worth stating. Count the decimal places in the two numbers being multiplied and add them.

The answer has that many decimal places. Here 0.3 and 0.4 have one each, so the answer has two.

The rule in full

Multiply as whole numbers, then count the decimal places in both numbers and put that many into the answer.

Nothing else is needed, and the rule never changes.

A worked product

Take 2.3 times 1.5. As whole numbers, 23 times 15 is 345.

Each number has one decimal place, so the answer has two. The answer is 3.45.

When one number is whole

Take 1.4 times 6. As whole numbers, 14 times 6 is 84.

The 1.4 has one decimal place and the 6 has none, so the answer has one. It is 8.4.

Three decimal places

Take 0.07 times 0.2. As whole numbers, 7 times 2 is 14.

The 0.07 has two decimal places and the 0.2 has one, so the answer needs three. Writing 14 with three places gives 0.014, and a zero has to be inserted to make room.

That zero is not optional

Notice what happened there. The digits 1 and 4 were not enough to fill three places, so a zero went in front of them.

Leaving it out would give 0.14, which is ten times too big. Always check that the answer has room for every place you counted.

Multiplying can make things smaller

The strip answer was 0.12, which is less than both 0.3 and 0.4. That is worth pausing over.

Multiplying by a number below 1 shrinks, exactly as it did with fractions. A tenth of something is smaller than the thing itself.

Trailing zeros may be dropped

Take 12.5 times 0.8. As whole numbers, 125 times 8 is 1000.

Two decimal places are needed, giving 10.00, and trailing zeros after the point add nothing. So the answer is simply 10.

An estimate as a check

Before trusting any answer, round both numbers and multiply those. For 2.3 times 1.5, round to 2 times 1.5, which is 3.

The answer 3.45 is close to 3, so it survives. Had it come out as 34.5, the estimate would have caught the misplaced point at once.

Which errors the estimate catches

An estimate will not catch a wrong digit, but it catches a point in the wrong place every time, because that changes the answer tenfold.

Since the point is what this topic is about, the estimate is the check worth making.

A second check on the places

The place count can be checked independently as well. Add the decimal places in the question and compare with the answer.

For 0.07 times 0.2 that is two plus one, so three places, and 0.014 has three. The two checks together are hard to slip past.

What carries forward

To multiply decimals, multiply as whole numbers and then count the decimal places in both numbers, giving the answer that many. Insert a zero if the digits do not fill the places needed.

Trailing zeros after the point may be dropped. Multiplying by a number below 1 makes the answer smaller, and an estimate from rounded numbers catches a misplaced point every time.

Scenario:

A shop sells floor strips by the square metre. A customer wants a strip 0.4 metres long and 0.3 metres wide, and an assistant answers 1.2 square metres, saying multiplying always makes things bigger. A second order is 2.3 metres by 1.5 metres, and the assistant writes 34.5. A third needs 0.07 metres by 0.2 metres, and he writes 0.14. A fourth is 12.5 metres by 0.8 metres, and he worries that his answer of 10.00 is missing something.

Analysis:

Three answers have the point in the wrong place or the wrong number of places, and the fourth is right but written oddly.

Activity:

Measure five rectangles to the nearest centimetre, work each area as a decimal product, and check every answer with a rounded estimate.

Title:

Count the places

Safety:

Nothing hazardous here.

What You Notice:

The shaded block holds 12 of the 100 small squares, which is 0.12, matching the rule. The other products come to 3.45, 8.4 and 0.014, and each rounded estimate lands close to its answer.

Each decimal place in a question is exactly one division by ten, which is the whole reason the place counts are added.

Leaving out the zero that makes room for a place produces an answer ten times too big, with nothing in it looking wrong.

A rounded estimate catches every misplaced point, because moving a point changes an answer tenfold.

Multiply the digits, then count the places.

Each place is one division by ten.

Sharing 4.8 metres among six

A rope of 4.8 metres is cut into 6 equal pieces. How long is each?

The sum is 4.8 divided by 6. The divisor here is a whole number, which makes this the easier kind.

Dividing by a whole number

Divide as you always have, and keep the decimal point where it was. Six goes into 48 eight times.

So 4.8 divided by 6 is 0.8 metres. The point stays above the point.

A second case of that kind

Take 9.1 divided by 7. Seven goes into 91 thirteen times.

Keeping the point in place gives 1.3. Nothing new was needed.

When the divisor is a decimal

Now take 0.36 divided by 0.4. Here the divisor itself carries a point, and long division has nothing to bite on.

So we change the question into one we can already do.

Multiply both numbers by ten

Multiply the 0.36 and the 0.4 by 10 each. That gives 3.6 divided by 4.

The divisor is now whole, so divide as before. Four goes into 36 nine times, giving 0.9.

Why that is allowed

This is the step worth understanding. A division is a fraction, with the first number on top and the second below.

Multiplying the top and bottom of a fraction by the same number leaves its value unchanged. So the new question has the same answer as the old one.

As many tens as you need

Take 7.5 divided by 0.25. The divisor has two decimal places, so multiply both numbers by 100.

That gives 750 divided by 25, which is 30. One multiplication cleared the divisor completely.

A third worked division

Take 0.6 divided by 0.03. Again the divisor has two decimal places, so multiply both by 100.

That gives 60 divided by 3, which is 20.

The answer can be larger

Notice those two answers. Dividing 7.5 by 0.25 gave 30, which is far bigger than the 7.5 we started with.

Dividing by a number below 1 makes the answer larger, exactly as it did with fractions. Small pieces fit in large numbers.

Why the divisor decides

It is the divisor that settles this, not the dividend. A divisor below 1 grows the answer, a divisor above 1 shrinks it, and a divisor of 1 changes nothing.

So 4.8 divided by 6 came out smaller, while 7.5 divided by 0.25 came out larger.

The check

Multiply your answer by the divisor and the dividend should come back. For 0.36 divided by 0.4 we answered 0.9.

Now 0.9 times 0.4 is 0.36, using the place count from the last topic. The answer stands.

Why that check is the right one

Division and multiplication undo each other, so the check tests the whole calculation rather than one step of it.

It catches a misplaced point, a wrong digit and a divisor cleared by the wrong number of tens.

An estimate first

An estimate is worth making too. For 7.5 divided by 0.25, note that 0.25 is a quarter, and there are 4 quarters in every whole.

So the answer should be about 4 times 7.5, which is 30. That is exactly what the working gave.

A common slip

The commonest error is multiplying only the divisor by ten and leaving the dividend alone. That changes the question.

If 0.36 divided by 0.4 is turned into 0.36 divided by 4, the answer comes out ten times too small. Both numbers must move together.

Which method for which question

So there is really one method with a first step. If the divisor is already whole, divide straight away.

If it is not, multiply both numbers by tens until it is, then divide. Nothing else changes.

It is worth noticing how little is new here. The division itself is the one you have always done, and the only addition is a first step that puts the question into that familiar form.

What carries forward

Dividing by a whole number needs no new work: divide as usual and keep the point in place. Dividing by a decimal needs the divisor cleared first, by multiplying BOTH numbers by ten as often as necessary.

That is allowed because a division is a fraction, and multiplying its top and bottom by the same number leaves its value alone. A divisor below 1 makes the answer larger, and multiplying the answer by the divisor checks the whole calculation.

Scenario:

A workshop has 4.8 metres of rope to cut into 6 equal pieces, and needs each length. It then has 0.36 of a litre of oil to pour into 0.4 litre cans, and an apprentice turns the sum into 0.36 divided by 4, answering 0.09. A third job divides 7.5 metres of tape into 0.25 metre strips, and the apprentice says the answer must be less than 7.5 because dividing always makes things smaller. He also asks how he could have checked any of them.

Analysis:

One division needs no clearing, one was cleared on one side only, and one rests on a rule that holds for some divisors and not others.

Activity:

Measure out five quantities, divide each by a decimal, and check every answer by multiplying it back by the divisor.

Title:

Clear the divisor

Safety:

Nothing hazardous here.

What You Notice:

Two of the five need no clearing and give 0.8 and 1.3. The others need 10, 100 and 100, giving 0.9, 30 and 20. Every answer multiplies back to its dividend, and the three that grew are exactly the three with a divisor below 1.

Clearing a divisor works only because a division is a fraction, whose top and bottom may be multiplied by the same number.

Dividing 7.5 by 0.25 gives 30, because every whole holds four quarters and 7.5 holds thirty of them.

Multiplying only the divisor by ten makes the answer ten times too small, and nothing in the working looks wrong.

Clear the divisor, then divide as usual.

Move both numbers, never just one.

Reading an answer before trusting it

A calculation can go wrong in a quiet way. A point slips one place and the answer still looks tidy, with every digit in it correct.

So it is worth knowing roughly where the answer should land before working it out exactly.

Rounding to the nearest whole

The simplest estimate rounds each number to the nearest whole number. The number 2.9 rounds to 3, and 4.1 rounds to 4.

Those are easy to work with, and the rounded answer is near the real one.

An estimated product

Take 2.9 times 4.1. Rounding gives 3 times 4, which is 12.

So the answer should be somewhere near 12. The exact value is 11.89, which it is.

How close is close enough?

An estimate is not meant to be the answer. It is meant to tell you which answers are possible.

Here 11.89 passes and 118.9 does not, and so does 1.189. The estimate rules out everything ten times out in either direction.

When both numbers round up

Rounding is not always balanced. Take 2.6 times 3.7, where 2.6 rounds up to 3 and 3.7 rounds up to 4.

The estimate is 12 again, but the exact answer is 9.62. Rounding both numbers up has pushed the estimate well above the truth.

So an estimate has a direction

That is worth knowing rather than ignoring. If both numbers were rounded up, the estimate is too high, and if both went down, it is too low.

With one of each, the two errors pull against each other and the estimate is usually closer.

An estimated quotient

Estimating works for division too. Take 20.4 divided by 5.1, and round to 20 divided by 5, which is 4.

The exact answer here is also 4, so the estimate happens to be spot on.

Estimating with a fraction in mind

Sometimes a fraction is the better tool. Take 0.48 times 0.52, where each number is close to a half.

A half of a half is a quarter, so the answer should be near 0.25. The exact value is 0.2496.

Where an estimate beats an exact answer

For some questions an estimate is all you want. Four items at 2.9 each comes to about 12, which tells you at once that a note of 20 will cover it.

The exact total is 11.6, and knowing it to the last penny changes nothing at the counter.

Judging a reported answer

The real use is judging somebody else's answer, including your own from five minutes ago. Ask whether it is near the estimate.

If a product of 2.9 and 4.1 is reported as 1.189, the estimate of 12 rejects it at once without any recalculation.

A tenfold error always shows

That works because a misplaced point multiplies or divides an answer by ten. No estimate is ever that far out.

So a point in the wrong place is caught every time, which is the error decimal work produces most.

A wrong digit may not show

An estimate is not a complete check, and pretending otherwise would be dishonest. If 2.9 times 4.1 is reported as 11.69, the estimate of 12 accepts it happily.

A single wrong digit usually moves an answer too little for an estimate to notice.

So use both kinds of check

The estimate catches the large errors and the exact check catches the small ones. For a product, recount the decimal places. For a division, multiply the answer back by the divisor.

Together they leave very little room, and each takes seconds.

Rounding to one figure

For large or awkward numbers, round to one significant figure instead of the nearest whole. That keeps the arithmetic to a single digit each.

The estimate is rougher, but a rough estimate still catches a tenfold error.

A habit rather than an extra step

Done properly, an estimate costs almost nothing. It is a glance before the work and a glance after it.

The point is to notice an impossible answer immediately, rather than at the end of a page of working.

What carries forward

Round each number and work out the rounded calculation to see roughly where the answer should land. Rounding both numbers the same way pushes the estimate that way too, so the direction is worth noting.

An estimate catches a misplaced point every time, since that is a tenfold error, but it will not catch a single wrong digit. So pair it with an exact check: the place count for a product, and multiplying back for a division.

Scenario:

A clerk checks four reported calculations. The first says 2.9 times 4.1 is 1.189. The second says 2.6 times 3.7 is 12, which the clerk accepts because his estimate was 12 as well. The third says 20.4 divided by 5.1 is 4, and he is unsure whether an estimate can confirm it. The fourth says 2.9 times 4.1 is 11.69, and his estimate of 12 accepts it. He concludes that estimating settles everything.

Analysis:

One answer is rejected rightly, one is accepted wrongly, one needs a different check, and the conclusion overstates what estimates do.

Activity:

Take ten reported calculations, estimate each before checking it exactly, and record which errors the estimate caught and which it missed.

Title:

Estimate, then check

Safety:

Nothing hazardous here.

What You Notice:

The estimates come out at 12, 12, 4 and 0.25 against exact answers of 11.89, 9.62, 4 and 0.2496. The one that is far out is the one where both numbers were rounded up, which is exactly what the direction note predicts.

An estimate rejects any answer ten times out in either direction, which is the commonest error in all decimal work.

An estimate that AGREES with a reported answer is no proof of it: for 2.6 times 3.7 both agree on 12 while the answer is 9.62.

Sometimes an estimate is all anyone wants: about 12 for four items at 2.9 each settles whether a note of 20 will do.

Round, work out, then compare.

An estimate rules out, it never rules in.