C1.4, C1.6; E1.4, E1.6 - Decimal calculations

C1.4, C1.6; E1.4, E1.6 - Decimal calculations

A bottle holds 0.60.6 litre and each glass holds 0.150.15 litre. How can dividing two numbers smaller than 1 give an answer greater than 1? This lesson connects decimal place value in C1.4 and E1.4 with signed addition, subtraction, multiplication and division in C1.6 and E1.6.

Decimal place value

The position of a digit tells you its value. Moving one place to the right makes a place worth one tenth as much: units, tenths, hundredths, thousandths. The decimal point separates the units from the tenths; it does not move when you add another zero at the end.

Decimal place value

A decimal records whole units and fractions with denominators 10, 100, 1000 and so on. For example, 0.4=4100.4=\frac4{10}, 0.04=41000.04=\frac4{100} and 0.004=410000.004=\frac4{1000}.

PlaceTensUnitsTenthsHundredthsThousandths
Value of one digit 1101110\frac1{10}1100\frac1{100}11000\frac1{1000}
Digits of 12.30512.30512305

The zero in 12.30512.305 holds the hundredths place. Without it, 12.3512.35 has five hundredths instead of five thousandths. By contrast, 12.305012.3050 has the same value as 12.30512.305: no extra amount has been added.

Worked example 1: reading decimal measures

Question. A length is 2.472.47 metres. Write it as whole metres plus tenths and hundredths of a metre. State the value of the digit 7.

Plan. Read each digit's position, then express its contribution as a fraction.

2.47=2+410+7100.2.47=2+\frac4{10}+\frac7{100}.

The 7 is two places after the decimal point, so it represents 7 hundredths.

Answer. 2+410+71002+\frac4{10}+\frac7{100} metres; the digit 7 has value 0.070.07 metre.

Check. 410=40100\frac4{10}=\frac{40}{100}, so the fractional parts total 47100=0.47\frac{47}{100}=0.47 metre.

The next measure has an empty tenths place. A zero keeps the hundredths from sliding into the wrong column.

Worked example 2: a placeholder zero

Question. A mass is 3 whole kilograms and 8 hundredths of a kilogram. Write it as a decimal. Write its fractional part as a fraction in its simplest form.

Plan. Put 0 in the tenths place and 8 in the hundredths place.

3+8100=3.088100=225divide numerator and denominator by 4\begin{aligned} 3+\frac8{100}&=3.08\\ \frac8{100}&=\frac2{25} &&\text{divide numerator and denominator by 4} \end{aligned}

Answer. 3.08 kg3.08\,\mathrm{kg}; the fractional part is 225\frac2{25} kilogram.

Check. 3.08 kg=3083.08\,\mathrm{kg}=308 hundredths of a kilogram, which is 300 hundredths plus 8 hundredths.

Multiplying a decimal by 10 makes each digit worth ten times as much; dividing by 10 makes it worth one tenth as much. Thus 0.38×10=3.80.38\times10=3.8 and 0.38÷10=0.0380.38\div10=0.038. Think about place values rather than simply removing zeros.

Watch out

0.80.8 and 0.080.08 are different: they are eight tenths and eight hundredths. But 0.8=0.800.8=0.80, because eight tenths is eighty hundredths.

Use the positions, including any empty places.

Addition and subtraction

Only equal-sized units can be added directly. Align decimal points so that tenths sit under tenths and hundredths under hundredths. Adding zeros at the right can make this clearer: 4.6=4.604.6=4.60.

For signed decimals, first decide what the signs mean. Adding a negative amount decreases the total; subtracting a negative amount increases it. For instance, a change of −1.2-1.2 degrees from 3.43.4 degrees gives 3.4+(−1.2)3.4+(-1.2).

Worked example 3: adding a negative change

Question. A temperature is 4.6∘C4.6^\circ\mathrm{C}. It changes by −1.85∘C-1.85^\circ\mathrm{C}. Work out the new temperature.

Plan. Add the signed change, then subtract aligned decimal places.

4.6+(−1.85)=4.60−1.85=2.75\begin{aligned} 4.6+(-1.85)&=4.60-1.85\\ &=2.75 \end{aligned}

In hundredths, this is 460−185=275460-185=275. The subtraction can regroup one whole unit as 100 hundredths when needed.

Answer. 2.75∘C2.75^\circ\mathrm{C}.

Check. Adding the fall back gives 2.75+1.85=4.602.75+1.85=4.60, the original temperature. The final temperature is lower, as a negative change requires.

The next subtraction removes a negative number. Change it to addition before calculating; do not ignore either sign.

Worked example 4: subtracting a negative decimal

Question. Work out −3.45−(−2.8)-3.45-(-2.8).

Plan. Subtracting a negative means adding its opposite; then compare the positive and negative amounts.

−3.45−(−2.8)=−3.45+2.80=−(3.45−2.80)the negative amount is larger=−0.65\begin{aligned} -3.45-(-2.8)&=-3.45+2.80\\ &=-(3.45-2.80) &&\text{the negative amount is larger}\\ &=-0.65 \end{aligned}

Answer. −0.65-0.65.

Check. Undo the subtraction: −0.65+(−2.80)=−3.45-0.65+(-2.80)=-3.45. On a number line, moving 2.8 right from −3.45-3.45 stops just left of zero.

The ordinary addition rules still apply. For two negative decimals, add their magnitudes and keep the negative sign: −0.7+(−0.26)=−0.96-0.7+(-0.26)=-0.96. A positive sum can carry ten hundredths into one tenth, exactly as whole-number addition carries ten units into one ten.

Watch out

Align place values, not the final digits. 3.7+0.48=3.70+0.48=4.183.7+0.48=3.70+0.48=4.18; treating 3.7 as 37 hundredths would change its value.

Decide the sign operation before subtracting.

Multiplication and division

For multiplication, place value explains the decimal point. Since 1.4=14101.4=\frac{14}{10} and 0.6=6100.6=\frac6{10}, their product is 14×6100\frac{14\times6}{100}. The whole-number product is divided by 100, not positioned by lining up decimal points.

For division, multiplying both dividend and divisor by the same non-zero number preserves the quotient. This lets you turn a decimal divisor into a whole number. For example, 4.2÷0.7=42÷74.2\div0.7=42\div7, because both quantities are ten times as large.

Worked example 5: a decimal product

Question. Work out 1.4×0.61.4\times0.6.

Plan. Multiply the whole-number digits, then account for the two factors of 10.

1.4×0.6=1410×610=84100=0.84\begin{aligned} 1.4\times0.6&=\frac{14}{10}\times\frac6{10}\\ &=\frac{84}{100}\\ &=0.84 \end{aligned}

Answer. 0.840.84.

Check. 0.60.6 of 1.41.4 is six tenths of it: 1.4÷10=0.141.4\div10=0.14, then 0.14×6=0.840.14\times6=0.84. The answer is less than 1.4, as expected.

The next calculation is division, with hundredths in the divisor. Scale both quantities by 100 so that the divisor is a whole number.

Worked example 6: a decimal divisor

Question. How many 0.150.15-litre glasses can be filled from a 0.60.6-litre bottle?

Plan. Divide total volume by one glass's volume, scaling both by 100.

0.6÷0.15=60÷15both amounts in hundredths of a litre=4\begin{aligned} 0.6\div0.15&=60\div15 &&\text{both amounts in hundredths of a litre}\\ &=4 \end{aligned}

Answer. 4 glasses.

Check. 4×0.15=0.604\times0.15=0.60 litre. The quotient counts glasses, so it can exceed 1 even though both volumes are less than 1 litre.

Use the same scaling for longer decimals, and keep the integer sign rules. For example, −0.24÷0.06=−24÷6=−4-0.24\div0.06=-24\div6=-4. A negative divided by a positive is negative; two negative signs would give a positive quotient.

Watch out

Scale both numbers in a division. 0.6÷0.15=60÷150.6\div0.15=60\div15, whereas 0.6÷150.6\div15 changes only the divisor and gives an answer 100 times too small.

Make the divisor a whole number, then multiply back to check.

Quick Recap

  • Place value distinguishes tenths, hundredths and thousandths; trailing zeros leave a decimal unchanged.
  • Align decimal points for addition and subtraction, and deal with negative signs deliberately.
  • Explain decimal products using powers of 10; scale both numbers in decimal division.

The opening bottle fills four glasses: its 60 hundredths of a litre contains four portions of 15 hundredths.