C1.4; E1.4 - Fractions, decimals and percentages

C1.4; E1.4 - Fractions, decimals and percentages

A container is 34\frac34 full. Another display says 0.750.75, and a third says 75%75\%: could all three describe the same amount? C1.4 and E1.4 use fractions, decimals and percentages as interchangeable ways of expressing a quantity, including quantities greater than one whole.

Fractions and decimals

A fraction is a division: 34=3÷4\frac34=3\div4. You can divide to find its decimal value, or build an equivalent fraction with denominator 10, 100 or 1000. In the other direction, read a terminating decimal by its place value, then simplify the fraction.

Equivalent forms

Equivalent forms represent exactly the same number. For example, 34=75100=0.75\frac34=\frac{75}{100}=0.75. Changing the form does not change the amount.

A decimal with three places is a number of thousandths: 0.125=12510000.125=\frac{125}{1000}. A zero in the middle still holds a place: 0.105=10510000.105=\frac{105}{1000}, not 15100\frac{15}{100}.

Worked example 1: fraction to decimal

Question. Write 58\frac58 as a decimal.

Plan. Make an equivalent fraction with denominator 1000, since 8×125=10008\times125=1000.

58=5×1258×125=6251000=0.625\begin{aligned} \frac58&=\frac{5\times125}{8\times125}\\ &=\frac{625}{1000}\\ &=0.625 \end{aligned}

Answer. 0.6250.625.

Check. 0.625×8=50.625\times8=5, so dividing 5 by 8 gives the stated decimal.

The next example reverses the direction. The number of decimal places chooses the denominator; simplification comes afterwards.

Worked example 2: decimal to simplest fraction

Question. Write 0.3750.375 as a fraction in its simplest form.

Plan. Write 375 thousandths, then divide numerator and denominator by their common factor 125.

0.375=3751000=38\begin{aligned} 0.375&=\frac{375}{1000}\\ &=\frac38 \end{aligned}

Answer. 38\frac38.

Check. 38+58=1\frac38+\frac58=1, while 0.375+0.625=10.375+0.625=1. The complementary amounts agree. Also, 3 and 8 have no common factor greater than 1.

Not every fraction gives a terminating decimal. Dividing 11 by 33 gives 0.333…0.333\ldots, with 3 repeating forever. A finite decimal such as 0.330.33 is an approximation, so write 13≈0.33\frac13\approx0.33, not an equality. Keep the fraction when an exact value is needed.

Watch out

Count every decimal place: 0.04=4100=1250.04=\frac4{100}=\frac1{25}, whereas 0.4=410=250.4=\frac4{10}=\frac25. The leading zero after the point matters.

Read the place value, then simplify.

Percentages

Percent means “out of one hundred”. A percentage compares an amount with a chosen whole. If 100 equal squares represent the whole, 36 squares represent 36100=0.36=36%\frac{36}{100}=0.36=36\%.

Percentage

p%p\% means p100\frac p{100}. One whole is 100%=1100\%=1. Percentages greater than 100%100\% represent more than one whole; for example, 150%=1.5150\%=1.5.

To change a decimal to a percentage, multiply its numerical value by 100 and attach %\%. To change a percentage to a decimal, divide its numerical value by 100. These actions are inverses: 0.36×100=360.36\times100=36, and 36÷100=0.3636\div100=0.36.

Worked example 3: less than one whole

Question. Express 0.360.36 as a percentage and as a fraction in its simplest form.

Plan. Read it as 36 hundredths, then use the meaning of percent and simplify the fraction.

0.36=36100=36%36100=925divide numerator and denominator by 4\begin{aligned} 0.36&=\frac{36}{100}=36\%\\ \frac{36}{100}&=\frac9{25} &&\text{divide numerator and denominator by 4} \end{aligned}

Answer. 36%36\% and 925\frac9{25}.

Check. 925×44=36100\frac9{25}\times\frac44=\frac{36}{100}, so both forms return to 0.360.36.

The next number has one whole as well as a decimal part. Convert the entire number, so the whole contributes 100%100\%.

Worked example 4: more than one whole

Question. Express 1.361.36 as a percentage and as a fraction in its simplest form.

Plan. Treat the whole as 100 hundredths and include the additional 36 hundredths.

1.36=136100=136%136100=3425divide numerator and denominator by 4\begin{aligned} 1.36&=\frac{136}{100}=136\%\\ \frac{136}{100}&=\frac{34}{25} &&\text{divide numerator and denominator by 4} \end{aligned}

Answer. 136%136\% and 3425\frac{34}{25}.

Check. 136%=100%+36%=1+0.36136\%=100\%+36\%=1+0.36. Also, 3425=1925=1.36\frac{34}{25}=1\frac9{25}=1.36.

The chosen whole matters. If a bottle's usual capacity is the reference, 120%120\% of that capacity is more than the bottle can hold; if a target number of points is the reference, scoring 120%120\% of the target is perfectly possible. A percentage has meaning only with its reference amount.

Watch out

5%=0.055\%=0.05, while 0.5=50%0.5=50\%. Write the division by 100 when changing a percentage to a decimal so that the decimal point is placed correctly.

Include all the hundredths, even when the amount exceeds one whole.

Equivalent forms

Choose a form that makes the comparison or calculation clear. A fraction shows a part of a stated total, a decimal is convenient in measurements, and a percentage compares amounts using a common hundred. Always identify the whole before forming the fraction.

If the ratio of an amount to its reference is ab\frac ab, divide aa by bb for the decimal and multiply that decimal by 100 for the percentage. This works when the amount is smaller or larger than its reference.

Worked example 5: expressing a part of a total

Question. A student answers 21 of 28 questions correctly. Express the proportion correct as a simplest fraction, a decimal and a percentage.

Plan. Use the number correct over the total, then convert equivalent forms.

2128=34simplify by 734=75100=0.750.75=75%\begin{aligned} \frac{21}{28}&=\frac34 &&\text{simplify by 7}\\ \frac34&=\frac{75}{100}=0.75\\ 0.75&=75\% \end{aligned}

Answer. 34\frac34, 0.750.75, 75%75\%.

Check. Three quarters of 28 is 28÷4×3=2128\div4\times3=21. The result should be below 100%100\%, since fewer than all 28 are correct.

The next amount exceeds its reference. The same conversion works, but the fraction is improper and the percentage exceeds 100.

Worked example 6: exceeding a target

Question. A club collects \54againstatargetofagainst a target of$40$. Express the amount collected as a simplest fraction, a decimal and a percentage of the target.

Plan. Use 40 as the whole, since the question compares the collection with the target.

5440=2720simplify by 22720=135100=1.351.35=135%\begin{aligned} \frac{54}{40}&=\frac{27}{20} &&\text{simplify by 2}\\ \frac{27}{20}&=\frac{135}{100}=1.35\\ 1.35&=135\% \end{aligned}

Answer. 2720\frac{27}{20}, 1.351.35, 135%135\% of the target.

Check. One whole target is \40;theextra; the extra $14isis\frac{14}{40}=0.35ofthetarget.Altogetherthatisof the target. Altogether that is1+0.35=1.35.Aroughestimate. A rough estimate 54\div40\approx1.4$ supports the size.

When a quotient has a recurring decimal, retain the exact fraction for an exact answer. If a question requests an approximation, use the requested accuracy and the symbol ≈\approx. A calculator can verify the division on a calculator paper; the equivalent-fraction method also works by hand.

Key idea

A percentage is another way to write a ratio to a whole. Changing form preserves the amount; changing the reference whole changes the comparison.

Start with the target as the denominator.

Quick Recap

  • A fraction is division; a terminating decimal can be written over 10, 100, 1000 and then simplified.
  • Percent means out of one hundred: 100%=1100\%=1, and percentages can exceed 100.
  • Identify the whole before converting a proportion in context.

All three opening displays describe the same amount: 34=0.75=75%\frac34=0.75=75\%.