Place value still matters below zero

1 · Learn

First decide the sign and which calculation the situation needs. For -12.45 + 3.78, adding a positive number moves towards zero. Because 12.45 is larger than 3.78, subtract the magnitudes and keep the negative sign. For -12.45 - 3.78, move further below zero: add the magnitudes and keep the negative sign.

Write decimal points underneath one another for addition or subtraction, so ones meet ones and hundredths meet hundredths. In 12.45 - 3.78, exchange a tenth to make 15 hundredths, then a one to make 13 tenths, then a ten to make 11 ones. Subtract column by column: 7 hundredths, 6 tenths, 8 ones. The magnitude is 8.67, so -12.45 + 3.78 = -8.67. Adding the magnitudes instead gives 16.23, so -12.45 - 3.78 = -16.23.

For (-12.45) × (-1.2), two negative factors give a positive result. Work out 1245 × 12 using two partial products: 2490 for ×2 and 12450 for ×10. Their sum is 14940. The original factors have three decimal places altogether, so divide this by 1000: 14.940 = 14.94. Do not simply line up decimal points in a multiplication.

For -12.45 ÷ 0.15, multiply BOTH values by 100: -1245 ÷ 15. In the positive written division, 15 goes into 124 eight times, using 120; the remainder is 4. Bring down the final 5 to make 45; 15 goes into 45 three times. Thus the magnitude is 83 and the signed quotient is -83. Check: -83 × 0.15 = -12.45.

Estimate the size before trusting an answer. Here -12 ÷ 0.15 is -80, so -83 is sensible; -0.83 is not. Division by a positive number less than one can increase the magnitude. Avoid a rule such as 'division always makes smaller'.

2 · Worked example

Calculate -8.64 ÷ (-0.12). Scale both by 100 to get -864 ÷ (-12). Two negatives give a positive quotient. 12 goes into 86 seven times (84), leaving 2; bring down 4, giving 24 and two more groups. Answer 72. Check 72 × (-0.12) = -8.64.

3 · Your turn

Using written methods, find -6.48 + 9.75, -6.48 - 9.75, (-6.48) × 1.25 and -6.48 ÷ (-0.18). Predict each sign first.

4 · Apply your learning

Use paper or the working pad to write each practice calculation in columns. Explain every exchange or partial product. Check subtraction with addition and division with multiplication; use a calculator only after recording your method.