Decimals & percentages

1Summary 重點摘要

Fractions, decimals and percentages are three outfits for the same number: 0.25 = 25100 = 25%. Learn to switch and discounts become easy.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. What a decimal isA decimal consists of an integer part and a fractional part separated by a decimal point, as in 3.14: the 3 is the integer part and .14 the decimal part. The first place after the point counts tenths and the second counts hundredths, so 3.14 = 3 + 110 + 4100. Place value matters most when comparing decimals: compare the integer parts first, then move right place by place — 3.2 beats 3.15 because its tenths digit 2 exceeds 1.
2. 2. What a percentage isA percentage is a special fraction with denominator 100, written with the % sign, meaning 'per hundred'. For example, 25% = 25100 = 0.25, 50% = 0.5, and 100% is the whole. Percentages express proportions and changes — discounts, growth, hit rates and interest are all everyday uses. Their advantage is a common base of 100, so any two can be compared directly: to compare 45% with 0.4, convert 0.4 into 40% and conclude that 45% is larger.
3. 3. Decimal to percentageTo convert a decimal into a percentage, multiply by 100%, which shifts the decimal point two places to the right and appends %. For example, 0.25 × 100% = 25%, 0.5 × 100% = 50%, and 0.075 × 100% = 7.5%. The principle is that 100% equals 1, so multiplying changes only the notation, not the value. A quick self-check: the number must get bigger; obtaining 2.5% from 0.25 signals a wrong direction and should be corrected at once.
4. 4. Percentage to decimalTo convert a percentage into a decimal, divide by 100%: drop the % sign and move the point two places left. For example, 75% = 0.75, 40% = 0.4, and 6% = 0.06 — note the extra zero, since 6% is 0.06 and not 0.6, the most common misplacement. This conversion is used constantly in calculations because percentages cannot be multiplied directly; to find 20% of 200, compute 200 × 0.2 = 40. Every discount and interest example in this lesson relies on it.
5. 5. Fraction to percentageConverting a fraction to a percentage takes two steps: divide numerator by denominator to get a decimal, then multiply by 100%. For 14: 1 ÷ 4 = 0.25, then 0.25 × 100% = 25%, so 14 = 25%; likewise 35 = 3 ÷ 5 = 0.6 = 60%. If the denominator divides 100, scaling directly is faster: 34 = 75100 = 75%. Memorise the common equivalents — 12 = 50%, 14 = 25%, 34 = 75%, 15 = 20%, 110 = 10% — for instant recall in examinations.
6. 6. Application: discountsDiscounts are the most common application. 八折 means paying 80% of the original price, that is, 20% off. A $200 book at 八折 costs 200 × 80% = 200 × 0.8 = $160. The worked example's trainers, originally $400, sell for 400 × 0.8 = $320, and the check looks at the saving: 400 − 320 = 80, with 80 ÷ 400 = 20%, matching the stated discount from the other direction. The universal formula is: sale price = original price × rate paid.
7. 7. Application: interestInterest is the other high-frequency application. Depositing $1000 at 5% per annum earns one year's interest of 1000 × 5% = 1000 × 0.05 = $50, and the total becomes 1000 + 50 = $1050. Distinguish carefully between interest and total: interest multiplies by the rate alone, while the total adds the principal. For two years at simple interest, the interest is 50 × 2 = $100. Converting the percentage to a decimal first, exactly as in discount questions, lets one routine handle both types.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• First after point: tenths; second: hundredths
• % means per 100
• Decimal to percentage: × 100%
• Percentage to decimal: ÷ 100%
• Discount: 20% off = 80% of original