Math Detours · Grade 5 · Sharpen · 1

Ten times bigger, ten times smaller

Every digit is worth ten times the one on its right. Slide the digits and you can multiply or divide by 10, 100 or 1000 without calculating.

Name ____________________Date ____________

Tonight's idea

A digit's value depends on its column. In 345.6 the 3 means three hundreds and the 6 means six tenths. A tenth is one of ten equal parts of 1.

Each column is worth ten times the column on its right. Ten ones make a ten. Ten tenths make a one. That is the only rule, and it never stops in either direction.

thousands hundreds tens ones tenths hundredths thousandths 3 4 5 6 3 4 5 6
34.56 × 10 = 345.6. Every digit slides one column left. The decimal point stays where it is.

So what does multiplying by 10 do? It makes every digit worth ten times as much. A digit worth ten times as much belongs one column to the left. So every digit slides one column left.

× 10 slides every digit one column left. ÷ 10 slides every digit one column right. For 100 slide two columns. For 1000 slide three.

Watch one: \(3.07 \times 100\) and \(45 \div 1000\)

\(3.07 \times 100\): slide each digit two columns left. The 3 lands in the hundreds, the 0 in the tens, the 7 in the ones. The answer is 307.

\(45 \div 1000\): slide each digit three columns right. The 4 lands in the hundredths and the 5 in the thousandths. Fill the empty columns with zeros so the digits keep their places: 0.045.

Watch the digits slide, and try your own numbers: mathdetours.com/demos/placevalue.html.

Your turn

  1. Multiply. (a) \(6.2 \times 10\)   (b) \(0.48 \times 100\)   (c) \(7 \times 1000\)   (d) \(1.305 \times 100\)
  2. Divide. (a) \(53 \div 10\)   (b) \(9.1 \div 100\)   (c) \(260 \div 1000\)   (d) \(4 \div 100\)
  3. What goes in the box? (a) \(0.72 \times \square = 72\)   (b) \(815 \div \square = 0.815\)   (c) \(\square \times 100 = 3.4\)   (d) \(\square \div 10 = 0.06\)
  4. (a) Sam says "to multiply by 10, write a zero on the end". Use 2.5 to show Sam what goes wrong. (b) In 5.55, how many times bigger is the value of the first 5 than the value of the last 5? (c) You multiply a number by 100 and then divide by 1000. Which single slide does the same job?

Stretch: a pile of paper

One sheet of paper is about 0.1 mm thick. How thick is a pile of 1000 sheets? How thick is a pile of a million sheets? Give the second answer in metres. There are 1000 mm in a metre.

Answers

  1. (a) 62. (b) 48. (c) 7000. (d) 130.5.
  2. (a) 5.3. (b) 0.091. (c) 0.26. (d) 0.04.
  3. (a) 100: the digits moved two columns left. (b) 1000: they moved three columns right. (c) 0.034. (d) 0.6.
  4. (a) Writing a zero gives 2.50, which is the same number as 2.5. Nothing slid. The right answer is 25. (b) 100 times. The first 5 is two columns to the left of the last, and each column is ten times bigger: \(10 \times 10 = 100\). (c) Two columns left, then three right, is one column right in all: divide by 10.

Stretch. \(0.1 \times 1000 = 100\) mm, which is 10 cm. A million is \(1000 \times 1000\), so slide three more columns: \(100\,000\) mm. Divide by 1000 to change to metres: 100 m. That is about the length of a soccer field.

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