Ganita Prakash II · Chapter 4 — Another Peek Beyond the Point

Decimals extend our Indian place-value system to the right of the ones place. Each place to the right is one-tenth of the place before it: tenths, hundredths, thousandths. So 27.53 means 2 tens, 7 ones, 5 tenths and 3 hundredths.

Fractions with denominators 10, 100, 1000. These turn straight into decimals. 254/1000 = 200/1000 + 50/1000 + 4/1000 = 2/10 + 5/100 + 4/1000 = 0.254. In the same way 7/10 = 0.7 and 35/100 = 0.35.

Grams to kilograms. Since 1 kg = 1000 g, 50 g = 50/1000 kg = 0.05 kg, 250 g = 0.25 kg and 25 g = 0.025 kg. When Jonali buys spices weighing 50 g, 100 g, 25 g and 250 g, she can write them as 0.05 kg, 0.1 kg, 0.025 kg and 0.25 kg.

A rule for dividing by 10, 100, 1000. To work out 123 ÷ 10:

• Write the number with a decimal point at its end: 123.
• Count the zeros in the divisor: 10 has one zero.
• Move the point that many places to the left, adding zeros in front if needed: 12.3.

So 24 ÷ 100 = 0.24, 678 ÷ 1000 = 0.678, 12 ÷ 1000 = 0.012 and 12345 ÷ 1000 = 12.345. The same rule works when the number already has a decimal point: 18.7 ÷ 100 = 0.187 and 18.7 ÷ 1000 = 0.0187.

Why it works. Dividing by 10 makes every digit worth one-tenth as much, so each digit moves one place to the right in the place-value chart. Moving the point to the left is the same thing seen from the digits’ side.

Multiplying by 10, 100, 1000 does the opposite: move the point to the right. So 5.7 × 100 = 570 and 0.306 × 1000 = 306.

In this chapter we build on these ideas to multiply and divide any decimals. You will see that the methods are the same as for whole numbers, with one extra step: placing the decimal point correctly.

Arshad buys 5 pens at ₹9.5 each. The cost is 9.5 × 5, which is 9.5 added five times: 47.5. But repeated adding is slow for something like 12.5 × 7.5. A better way is to write each decimal as a fraction.

The idea. 9.5 = 95/10. To multiply fractions, multiply the tops and multiply the bottoms: 95/10 × 5/1 = 475/10 = 47.5. So the pens cost ₹47.50.

A car. A car runs 12.5 km on one litre of petrol. With 7.5 litres it covers 12.5 × 7.5 = 125/10 × 75/10 = 9375/100 = 93.75 km.

Walking to school. Ajay's school is 827 m, or 0.827 km, from home. He walks there and back, 6 days a week. In a day he walks 0.827 × 2 = 1654/1000 = 1.654 km. In a week, 1.654 × 6 = 9924/1000 = 9.924 km.

Area. A rectangle 5.7 cm by 13.3 cm has area 57/10 × 133/10 = 7581/100 = 75.81 square cm.

Look at the denominators. Each decimal becomes a fraction over 10, 100 or 1000. Multiplying these denominators gives another 1 followed by zeros, and the number of zeros adds up: 10 × 10 = 100, 10 × 100 = 1000. This is why the answer can always be written straight back as a decimal.

Can the product be a whole number? Yes. 2.5 × 0.4 = 25/10 × 4/10 = 100/100 = 1, and 0.5 × 4 = 2. A decimal times a decimal, or a decimal times a whole number, can give a whole number.

The fraction method shows a pattern. Count the digits after the decimal point in each number and in the answer:

The places in the answer are always the sum of the places in the two numbers. This is because the denominators 10, 100, 1000 multiply, and their zeros add up.

The rule.

• Remove the decimal points and multiply the whole numbers.
• Add up the number of decimal places in the two numbers.
• Place the point in the product so that it has that many decimal places, counting from the right.

Example. 5.8 × 1.24. First 58 × 124 = 7192. There are 1 + 2 = 3 decimal places, so 5.8 × 1.24 = 7.192. Check by estimating: about 6 × 1.2 = 7.2, so 7.192 is sensible.

Using a known product. If 596 × 248 = 147808, then 5.96 × 24.8 has 2 + 1 = 3 places, so it is 147.808. Similarly, from 18 × 12 = 216: 18 × 1.2 = 21.6, 1.8 × 1.2 = 2.16, 0.18 × 0.12 = 0.0216 and 0.018 × 0.012 = 0.000216.

Zeros at the end. 56.50 and 56.5 are the same number, and so are 2.250 and 2.25. You may drop such zeros before multiplying; you will get the same product, 127.125, for the price of 2.25 kg of oranges at ₹56.50 per kg.

Filling places with zeros. 0.432 × 0.23: 432 × 23 = 9936 and we need 3 + 2 = 5 places, so write 0.09936, putting a zero in front.

Always estimate. A quick estimate catches a misplaced point. 27.34 × 6 is about 27 × 6 = 162, so the answer 164.04 is right and 16.404 is not.

With counting numbers, multiplying never makes a number smaller: 5 × 3 = 15 is bigger than both. With decimals this is no longer always true. Compare:

Why? Multiplying by a number less than 1 means taking only a part. 0.25 × 8 is a quarter of 8, which is 2, less than 8. And 0.25 × 0.8 is a quarter of 0.8, which is even smaller than 0.8.

Three situations. Decimals behave just like fractions here.

Predicting without calculating. Which of these products is less than 1: 7 × 0.6, 0.7 × 0.6, 0.7 × 6, 0.07 × 0.06? The products with both numbers less than 1 must be less than 1, so 0.7 × 0.6 = 0.42 and 0.07 × 0.06 = 0.0042 are. The others are 4.2 each, more than 1.

Multiplying by exactly 1 leaves a number unchanged, so 1 is the dividing line. A factor above 1 stretches the other number; a factor below 1 shrinks it.

Using this as a check. If you work out 0.8 × 45 and get more than 45, you have made a mistake, because 0.8 is less than 1. Thinking ‘stretch or shrink’ before calculating is a quick way to catch errors.

Everyday meaning. 0.5 × ₹240 is half of ₹240, which is ₹120. 1.5 × ₹240 is one and a half times ₹240, which is ₹360.

Anuja cuts a 3.9 m ribbon into 10 equal pieces. Each piece is 3.9 ÷ 10 m. As a fraction, 3.9 = 39/10, and dividing by 10 means multiplying by 1/10: 39/10 × 1/10 = 39/100 = 0.39 m. Cut into 100 pieces, each would be 39/1000 = 0.039 m, which is 3.9 cm or 39 mm. This agrees with the rule of moving the point to the left.

Sharing a whole number. Neenu shares 29 m of ribbon equally with Anu. Each gets 29 ÷ 2 m: 14 m each with 1 m left, and that 1 m split in two gives ½ m more. To turn ½ into a decimal, find an equal fraction with denominator 10: ½ = 5/10 = 0.5. So each gets 14.5 m.

Sharing among four. Now share the 29 m among 4 friends: each gets 29/4 m. Is 4 a factor of 10? No. Of 100? Yes, 4 × 25 = 100. So 29/4 = (29 × 25)/(4 × 25) = 725/100 = 7.25 m.

The method. Write the division as a fraction, then multiply the top and bottom by a number that makes the bottom 10, 100 or 1000.

When does this work? It works when the denominator has only 2s and 5s as prime factors, because 10 = 2 × 5, 100 = 2 × 2 × 5 × 5, and so on. Notice the pattern: 1/2 = 0.5, 1/4 = 0.25, 1/8 = 0.125, 1/16 = 0.0625, while 1/5 = 0.2, 1/25 = 0.04, 1/125 = 0.008.

For a fraction such as 10/3, no multiplier turns 3 into 10, 100 or 1000, since 3 is not a factor of any of them. For such divisions we need a more general method: long division using place value, which comes next.

Long division divides one place at a time, from the largest place downwards, regrouping what is left into the next smaller place. For 1324 ÷ 4: 1 thousand cannot be shared by 4, so regroup it as 10 hundreds; 13 hundreds ÷ 4 = 3 hundreds, 1 left; 12 tens ÷ 4 = 3 tens; 4 ones ÷ 4 = 1. So 1324 ÷ 4 = 331.

Going past the ones. Try 1325 ÷ 4. As before we reach 331 with 1 one left over. Instead of stopping with a remainder, regroup 1 one as 10 tenths. 10 tenths ÷ 4 = 2 tenths, with 2 tenths left. Regroup these as 20 hundredths: 20 ÷ 4 = 5 hundredths, nothing left. So 1325 ÷ 4 = 331.25.

The key step. The moment we regroup ones into tenths, we put a decimal point in the quotient. After that, write zeros after the point in the dividend whenever more places are needed.

Example: 237 ÷ 8.

So 237 ÷ 8 = 29.625. Check: 29.625 × 8 = 237.

A decimal dividend. A shopkeeper packs 9.5 kg of sugar equally into 4 bags. 9 ones ÷ 4 = 2, 1 one left. Put the point. 15 tenths ÷ 4 = 3, 3 tenths left. 30 hundredths ÷ 4 = 7, 2 left. 20 thousandths ÷ 4 = 5. Each bag holds 2.375 kg.

A small dividend. 0.06 ÷ 5: 0 ones, then point, 0 tenths, 6 hundredths ÷ 5 = 1 with 1 left, 10 thousandths ÷ 5 = 2. So 0.06 ÷ 5 = 0.012. Do not forget the zero in the tenths place.

Checking with fractions. 1325/4 = (1325 × 25)/(4 × 25) = 33125/100 = 331.25, the same answer.

Ravi rides 126 km from Pune to Matheran in 2.5 hours. His average speed is 126 ÷ 2.5 km per hour. How do we divide by a decimal?

Make the divisor a whole number. Multiplying both the dividend and the divisor by the same number does not change the quotient, just as 6 ÷ 3 and 60 ÷ 30 are both 2. So multiply both by 10: 126 ÷ 2.5 = 1260 ÷ 25. Long division gives 50.4. Ravi's average speed is 50.4 km per hour.

More examples. 4.68 ÷ 1.3: multiply both by 10 to get 46.8 ÷ 13 = 3.6. And 4.68 ÷ 0.13: multiply both by 100 to get 468 ÷ 13 = 36.

How many places to move? Count the decimal places in the divisor only, and move the point that many places right in both numbers. The dividend may still be a decimal afterwards; that is fine, since we already know how to divide a decimal by a whole number.

Can the quotient be bigger than the dividend? With whole numbers, 128 ÷ 4 = 32 is smaller than 128. But 128 ÷ 0.4 = 320 is bigger! Dividing by 0.4 asks how many pieces of 0.4 fit into 128, and many small pieces fit.

Using a known fact. From 156 × 12 = 1872, we get 187.2 ÷ 1.2 = 156 and 18.72 ÷ 15.6 = 1.2, by counting places just as in multiplication.

Try 10 ÷ 3 by long division. 10 ones ÷ 3 = 3, 1 left. 10 tenths ÷ 3 = 3, 1 left. 10 hundredths ÷ 3 = 3, 1 left. The remainder is 1 every time, so the same step repeats for ever: 10 ÷ 3 = 3.333… The dots mean the 3s never stop. Such a quotient cannot be written with a finite number of digits.

This happens because 3 is not a factor of 10, 100, 1000 or any such number. Similarly 10 ÷ 9 = 1.111… and 100 ÷ 11 = 9.0909…

A remainder cycle. Divide 1 by 7 and watch the remainders: 1, 3, 2, 6, 4, 5, then 1 again. Once a remainder repeats, every step after it repeats too, so the digits of the quotient repeat: 1 ÷ 7 = 0.142857 142857 … When dividing by 7 the only possible non-zero remainders are 1 to 6, so a repeat must come within six steps.

The magic number 142857. Multiply it by 1 to 6:

The first six products use the same digits in the same cyclic order, just starting at a different place. Multiplying by 7 gives 999999. Numbers with this property are called cyclic numbers. Another one comes from the repeating block of 1 ÷ 17. In 1927 the mathematician Emil Artin conjectured that there are infinitely many cyclic numbers. Nobody has proved or disproved this yet.

Which fractions end? A fraction in lowest terms gives an ending decimal exactly when its denominator has no prime factors other than 2 and 5. So 3/8 and 7/20 end, but 1/3, 1/6 and 1/7 go on for ever.

In real life we round such answers. Sharing ₹10 among 3 people gives ₹3.33 each, with 1 paisa left over.

The Earth takes about 365.2422 days to go round the Sun, but a calendar year has 365 days. After one year the Earth still needs 0.2422 of a day to finish its circle. That seems tiny, but after 100 years it adds up to 0.2422 × 100 = 24.22 days, and the seasons would drift away from the calendar months.

First fix: a leap day every 4 years. Years divisible by 4 get 366 days. In 4 years the calendar has 4 × 365 + 1 = 1461 days, while the Earth needs 4 × 365.2422 = 1460.9688 days. Very close. In 100 years: 25 years are divisible by 4, so the calendar has 100 × 365 + 25 = 36,525 days, but the Earth needs 100 × 365.2422 = 36,524.22 days. Now the calendar is ahead by 0.78 of a day. We have overcorrected.

Second fix: skip the leap day in century years. A year divisible by 100 is not a leap year. In 100 years that leaves 24 leap years and 76 ordinary ones: 24 × 366 + 76 × 365 = 36,524 days, just 0.22 days short. Over 1000 years this becomes 365,240 days against the Earth's 365,242.2, a gap of 2.2 days.

Third fix: every 400th year is a leap year after all.

Total: 750 × 365 + 240 × 366 + 8 × 365 + 2 × 366 = 365,242 days. The Earth needs 365,242.2 days, so the calendar is short by only 0.2 of a day in 1000 years. This is the rule our Gregorian calendar uses. So 2000 was a leap year, 1900 was not, and 2024 was.

The lesson. A small decimal, multiplied by a large number, can become large. Decimal multiplication helps us check how good a rule really is.

A shopping check. A 210 g packet of chikki costs ₹70.5 and a 110 g packet of chips costs ₹33.25. Per gram, chikki costs about ₹0.336 and chips about ₹0.302, so chips are cheaper per gram.