Place Value and Ordering Numbers

Identify the column containing the digit 8.

The first column after the decimal point is tenths.

Use a zero in the hundreds column.

Keep a column for every place value, including the zeros.

Each digit becomes one thousand times its previous value.

Dividing by 1000 moves each digit three place-value columns to the right.

1. The 5 moves from hundreds to tenths. 508\div1000=0.508.

Compare the digits in matching columns.

1. Write 0.68 as 0.680. The tenths agree, but 8 hundredths is greater than 0 hundredths, so 0.68 is greater.

Compare tenths first, then hundredths, then thousandths.

Part (a)

1. The order is 0.067, 0.076, 0.700, 0.706, 0.760. The second number is 0.076.

Part (b)

1. The fourth number is 0.706. It is greater than 0.700 and less than 0.760.

Count the gaps between two labelled ticks.

1. From 2.4 to 2.6 there are four equal gaps. Each gap is 0.2\div4=0.05.
2. A is one gap after 2.6, so its value is 2.65.

On a number line, the number farther left is smaller.

1. Write -1.4 as -1.40. It is farther below zero than -1.04, so -1.4<-1.04.

Use the same unit before comparing.

1. 0.45 kg is 450 g and 0.405 kg is 405 g.
2. Comparing 450, 460 and 405 gives the lightest mass as 405 g.

Compare the distance from each number to 3.1.

1. The distances from 3.1 are 0.02, 0.7, 0.092 and 0.78, respectively.
2. The smallest distance is 0.02, so 3.08 is closest.

Express both numbers in hundredths.

1. 0.43 is 43 hundredths; 0.5 is 50 hundredths. Therefore 0.43<0.5.
2. The number of digits alone does not tell you which decimal is greater.

Numbers farther below zero come first.

1. The order is -3.400, -3.040, -3.004, -0.340, -0.304.
2. The second number is -3.04.

Compare the place value of the digit 8 before and after.

1. The 8 moves from hundredths to tens: three columns to the left. The multiplier is 1000.

Choose the smallest possible digit in each column, starting from the left.

1. The thousands digit must be 5 or 8. Choose 5 to make the number smaller.
2. Put the remaining digits in ascending order: 0, 2, 8. The number is 5028.

Find the step size before reading the points.

Part (a)

1. There are eight gaps. Each gap is 0.4\div8=0.05.
2. A is three gaps after 1.2: 1.2+3\times0.05=1.35.

Part (b)

1. There are four gaps from A to B. The difference is 4\times0.05=0.20.

Count up from 4.05 in steps of 0.01. Check whether the two given numbers can be included.

1. Going up in hundredths from 4.05 gives 4.06, 4.07, 4.08, 4.09, 4.10.
2. The number must be greater than 4.05 and less than 4.10, so neither 4.05 nor 4.10 is allowed.
3. The possible values are 4.06, 4.07, 4.08 and 4.09.

First work out which tenths digit is possible.

1. Every number with two decimal places that is greater than 7.40 and less than 7.50 has tenths digit 4.
2. Twice 4 is 8, so the hundredths digit is 8. The number is 7.48.

A tens digit changes the difference more than a units digit does.

1. Make the larger tens digit as large as possible and the smaller tens digit as small as possible: use 9 and 1.
2. Use 7 in the larger units position and 3 in the smaller one. The greatest difference is 97-13=84.