3.2 Fractions, Decimals, Percentages & Percentage Change

Fractions, decimals, percentages and percentage change

One quantity, three forms

A fraction, decimal, and percentage can express the same proportion:

3/4 = 0.75 = 75%

Convert a fraction to a decimal by dividing numerator by denominator. Convert a decimal to a percentage by multiplying by 100. Convert a percentage to a decimal by dividing by 100. Simplify fractions by dividing numerator and denominator by a common factor.

Memorise useful benchmarks through understanding:

These anchors speed estimates. Seven eighths must be greater than three quarters, so its decimal must exceed 0.75.

Fraction operations

For addition or subtraction, use a common denominator:

2/3 + 1/4 = 8/12 + 3/12 = 11/12.

For multiplication, multiply numerators and denominators, cancelling common factors first where convenient:

3/5 × 10/9 = 30/45 = 2/3.

Dividing by a fraction means multiplying by its reciprocal:

3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8.

Do not add denominators directly. The denominator tells the size of the parts; parts must be expressed in the same size before they are combined.

Mixed numbers should be converted deliberately. For multiplication or division, rewrite 2 1/3 as 7/3. For addition, you may combine whole and fractional parts separately when the fractions are easy. An improper fraction is not an error; convert it to a mixed number only if the requested answer format or interpretation benefits.

When comparing fractions with unlike denominators, use a common denominator, cross-products, or decimals. For positive denominators, 5/8 versus 3/5 can be compared through 5×5 = 25 and 3×8 = 24, so 5/8 is slightly larger.

Decimal place value

Align decimal points for addition and subtraction. For multiplication, multiply as whole numbers and then place the decimal according to the total decimal places. For division, move the decimal in both dividend and divisor by the same number of places until the divisor is whole.

Example: 7.2 ÷ 0.6 becomes 72 ÷ 6 = 12. A quick estimate confirms that twelve lots of six tenths equal 7.2.

Find a percentage of an amount

Use:

percentage amount = decimal percentage × base amount.

Thirty-five percent of 240 is 0.35 × 240 = 84. Mentally, 30% is 72 and 5% is 12, totalling 84.

For “84 is what percentage of 240?”, divide part by whole:

84 ÷ 240 × 100% = 35%.

The challenge is often identifying the whole. In “18 of 60 appointments were changed,” 60 is the base because all appointments form the reference set.

Percentage increase and decrease

For a change from old to new:

percentage change = (new − old) ÷ old × 100%.

A value rising from 80 to 92 changes by 12. The percentage increase is 12 ÷ 80 = 15%. Dividing by 92 would answer a different question: the change as a share of the new value.

Use multipliers:

• increase by r%: multiply by 1 + r/100;
• decrease by r%: multiply by 1 − r/100.

A 15% increase on 80 is 80 × 1.15 = 92. A 15% decrease from 92 is 92 × 0.85 = 78.2, not 80. Equal percentage increases and decreases do not cancel because the second percentage uses a different base.

Successive changes

Apply factors in sequence. A value increased by 20% and then decreased by 10% becomes:

original × 1.20 × 0.90 = original × 1.08.

The net result is an 8% increase. Adding +20 and −10 happens to suggest +10%, which is wrong because the decrease acts on the enlarged value.

Reverse percentages

If a final value includes a change, divide by the multiplier. If $276 is the price after a 15% increase, original × 1.15 = 276, so original = 276 ÷ 1.15 = 240.

Do not subtract 15% of the final value; that uses the wrong base. Reverse calculations undo the multiplier.

Percentage points versus percent

If a rate rises from 40% to 50%, it increases by 10 percentage points. Relative to the original 40%, that is a 25% increase because 10 ÷ 40 = 0.25. Questions may test this distinction.

Reasonableness checks

For a positive base, 10% is one tenth, 1% is one hundredth, and 50% is half. If 35% of 240 is reported as 840, the result exceeds the whole and is impossible. Check whether the wording asks for the changed amount or the new total: “increase of 15%” is the extra part; “increase by 15%” often asks for the final value.