Decimals MTH 3 03a, MTH 3 03b, MTH 3 0a, MNU 3-0a, MNU 3-08a Place Value It is vital to understand place value in decimal fractions. The decimal point is always between the units (ones) and the tenths column. We have: Thousands Hundreds Tens Units Tenths Hundredths Thousandths 2 5 7 8 9 3 In the number 257.893, the 7 is worth 7 tens while the 8 is worth 8 tenths. Add and Subtract decimals Adding decimals Set out your calculations by putting tenths, hundredths, thousandths etc below each other. Add trailing zeros to ensure all numbers have the same number of digits after the decimal point. (a) 6.456 + 7.78 (b).8 + 3.785 + 55.243 6.456.800 + 7.780 3.785 24.236 + 55.243 60.828 Subtracting decimals Set out your calculations by putting tenths, hundredths, thousandths etc below each other. Add trailing zeros to ensure all numbers have the same number of digits after the decimal point. (a) 9.475 5.6 (b) 9 3.654 9 9 8 9. 475 8 9. 0 0 0-5. 600-3. 6 5 4 3. 875 5. 3 4 6 When adding or subtracting decimals it is important that we line up the columns so that the place value is correct (i.e. we line up the decimal points) So, 2.7 + 83.62 becomes: 2.7 + 83.62 but it is better to have the same amount of decimal places in each number so we write:
2.70 + 83.62 This is particularly important in subtraction where 5 2.7 becomes: 5.00-2.7 We also may have applications of decimal addition and subtraction in contexts such as time or money. For instance: John ran a race in a time of 54.6 seconds. 6 Terry ran second faster. What was Terry s time? 00 54.60-0.06 54.54 secs 6 sec = 0.06 sec. He ran faster to so took less time. 00 i.e. subtract 0.06 sec.. State the value of the 5 in each of the following: a) 24.56 b) 509.2 c) 0.0567 d) 532.6 e) 2.05 2. Set out and evaluate: a) 6.84 + 2.39 b) 6.8 + 5.47 c) 3..87 d) 4 2.63 e) 5.8 add 0 2 f) 4. - 0 6 8 g) 5.8 + 00 9 h) 3.4-000 3. Gary ran a race in 38.2 secs. 7 a) If Bob ran secs faster, what was his time? 00 8 b) If Neil ran secs slower, what was his time? 0 Ordering Decimals So that we can compare the value of different decimals, it is important that we have all of the numbers in the same format e.g. all with 3 decimal places. For example, place these in order, starting with the smallest: 0.03 0.3 0.30 0.03 0.3 0.003 0.0 Writing each with 3 decimal places gives: 0.030 0.300 0.30 0.03 0.30 0.003 0.00
Now we can compare each value so in order we have: 0.003 0.00 0.03 0.030 0.300 0.30 0.30 Write each of these lists in order of size, starting with the smallest: a) 0.5 0.25 0.02 0.005 0.52 0.002 b).6.006.6.06..06 c) 5.32 5.02 5.03 5.3 5.2 5.32 Multiplying and Dividing by 0, 00, 00 When we multiply a number by 0, the value of each digit increases by a factor of 0. The result of this is that the digits move left one place as shown below: 0 00 6 5 7 6 5 7 000 So 65.7 x 0 = 657. Again, 0.24 x 0 becomes 2.4 as below. 0 2 4 2 4 Equally, multiplying by 00 moves digits 2 places to the left: 0 00 000 0 5 8 5 8 0 00 000 So 5.8 x 00 = 580 Here there is a need to fill in the units column with a zero to maintain the other digits in the correct place. And also multiplying by 000 will move the digits 3 places to the left, hence: 0 00 0 0 4 4 0 000
So 0.04 x 000 = 40 (again there is the need to fill in the units column with a zero in this case). Dividing by 0, 00 and 000 works in a similar manner, with the numbers this time getting smaller and so moving to the right. 0 00 2 8 0 2 8 000 So 2.8 0 = 0.28. Evaluate each of the following multiplications: a) 0.32 x 0 b) 2.4 x 00 c) 3. x 000 d) 0.04 x 0 e) 0.092 x 00 f) 0.0862 x 000 g) 25. x 00 h) 27 x 0 2. Evaluate each of the following divisions: a) 32.6 0 b) 28 00 c) 56.3 00 d) 26.5 000 e) 94.56 000 f) 0.02 0 g) 25400 00 h) 3.6 000 Multiplying and Dividing Decimals This works in a similar manner to multiplication and division of whole numbers. It is important to keep the digits lined up with the decimal point in place, and to remember to include the decimal point in the answer. e.g. A roll of wallpaper is 2.4 metres long. Find the length of 7 rolls. 2. 4 x 7 6. 8 metres e.g. 2 One fifth of the weight in a packet of crisps is fat content. If I eat a large portion of crisps weighing 43.25 grams, how many grams of fat have I eaten? 8.65 5 43.25 Multiplying by Multiples of 0, 00, 000 To multiply by a multiple of 0 we split the calculation up into two portions, just as we did with whole numbers.
e.g. 6.2 x 200 becomes 6.2 x 2 x 00 = 2.24 x 00 = 224 Remember we can rearrange the order of our multiplication always split up the multiple of 0. e.g. 2 3000 x 5.07 becomes 5.07 x 3 x 000 = 5.2 x 000 = 520. Evaluate: a) 2.3 x 200 b).4 x 50 c) 7.04 x 6000 d) 400 x 0.28 e) 200 x 78. Dividing by Multiples of 0, 00, 000 As with multiplying, we split up the number we are dividing by. e.g. 24.8 200 becomes 24.8 2 00 = 2.4 00 = 0.24 e.g. 2.605 50 becomes.605 5 0 = 0.32 0 = 0.032. Evaluate: a) 50.2 20 b) 3.2 300 c) 8.545 50 d) 264 3000 e) 40.4 60 Rounding and Estimating To round a number to decimal place the answer must have exactly one figure to the right of the decimal point. To decide if the number is closer to the value above or below, we need to look at the value of the hundredths column (the second decimal place). e.g. Round to d.p: 5.872 5.9 Look at this number 7 is more than halfway so we round up. 5.849 5.8 Look at this number 4 is less than halfway so we round to the value below. Note: if the next number is a 5 we always round up.
The same process applies for rounding to 2 or 3 decimal places as shown: Round 3.567 to 2 d.p. Round 3.998 to 3 dp 3.567 3.998 3.57 3.200. Round these values to decimal place: a) 6.3 b) 4.25 c) 6.762 d) 2.209 e) 3.986 2. Round these values to 2 decimal places: a) 2.30 b) 5.692 c) 4.225 d) 5.6957 e) 2.003 3. Round these values to 3 decimal places: a) 3.246 b) 3.69 c) 4.0565 d) 29.9999 e).333333 Foreign Currency To exchange into an amount of money into a foreign currency we multiply by the exchange rate. To exchange into an amount of money out of a foreign currency we divide by the exchange rate. e.g. e.g. 2 The exchange rate is = 4.3 Polish zloty. How many zlotys will I get in exchange for 200. Number of zlotys = 4.3 x 200 = 4.3 x 2 x 00 = 8.6x 00 = 860 zloty The exchange rate is = 300 Hungarian Forints. How many pounds will I get if I exchange 33600 Forints? Number of pounds = 33600 300 = 33600 3 00 = 200 00 = 2