Percent (I): introduction

1Summary 重點摘要

1% = 1100. 50% = half.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Part 1: What % meansTake grid paper with exactly 100 small squares and shade 25 of them: the shaded part is 25%, read as twenty-five percent. A percentage has its denominator fixed at 100, meaning 'how many out of every hundred'. Why fix the standard at 100? Because with the same standard, different percentages can be compared directly: 25% against 40% shows at once that 40% is larger, with no need to convert fractions. For example, two classes of 50 and 40 pupils send 30 and 20 on a trip; as percentages these are 60% and 50%, and one glance shows the first class joined in a larger share. A common mistake is to treat a percentage as a fixed quantity; a percentage expresses only a ratio, and one must always say 'percent of what'.
2. Part 2: How to read percentages25% reads twenty-five percent, 7% reads seven percent, and 100% reads one hundred percent; the sign % stands for 'out of every hundred'. Read the number first and then the word percent, the same order as reading a fraction, because 25% is 25100 to begin with. Why practise reading? Because daily information arrives by listening and reading, such as a 70% chance of rain in the weather report or 80% battery on the phone, and only correct reading gives correct understanding. While practising, rewrite the % into fraction form and then read it out. A common mistake is to reverse the order, reading 25% as 'one hundredth of twenty-five', or to think % may stand before the number; % is always written after the number, and one hundred percent is written 100%.
3. Part 3: Percent to fractionChanging a percentage into a fraction takes just two steps: first drop the % and write a fraction with denominator 100; second simplify. For example, write 25% as 25100, divide the top and the bottom by 25, and it simplifies to 14; in the same way 50% = 50100 = 12 and 75% = 75100 = 34. Why is this allowed? Because the very definition of a percentage is 'out of every hundred', so the denominator 100 is part of its meaning, not an added trick. The simplified fraction has exactly the same value as the original percentage, only written differently: 14, 25100 and 25% all name the same amount. A common mistake is to skip simplifying and leave 25100 as the final answer, or to simplify wrongly, turning 50100 into 15; the top and the bottom must be divided by the same number.
4. Part 4: Percent to decimalTo change a percentage into a decimal, divide the number by 100 — in other words, move the decimal point two places left: 25% = 25 ÷ 100 = 0.25; 7% = 7 ÷ 100 = 0.07; 120% = 1.2. Why two places left? Because dividing by 100 shrinks a number to one hundredth of itself, and each place the point moves left shrinks it ten times, so two places give one hundredth. If 7% were written as 0.7 it would be ten times too big, since 0.7 is really 70%. The other way round, a decimal becomes a percentage by moving the point two places right and adding %: 0.45 = 45% and 0.07 = 7%. A common mistake is to move the point only one place, writing 7% as 0.7 or 3% as 0.3; after moving, count the places and check the direction with a question like 'should 7% be less than 0.1?'
5. Part 5: Three special valuesThree special values must be learned first: 100% means the whole, so eating the entire cake is eating 100% of it; 50% means half, so a half cup is 50% of a full cup; 0% means none at all. Scoring 90 out of a full 100 marks gives 90%, and scoring 100 out of a full 200 also gives 50%. Why are these three values the most important? Because they are the yardstick for judging all other percentages: above 100% means more than the whole, such as output rising to 120% of the original, and whether something passes half is seen by comparing with 50%. A common mistake is to think percentages are always below 100%, but growth can exceed it: a price rising from 10 dollars to 25 dollars is a 150% rise. Another is taking 50% as 150, whereas 50% equals 12.
6. Part 6: Percentages in daily lifePercentages appear everywhere in daily life: a phone showing 80% battery has four fifths of its charge left; a shop discount of one tenth means paying 90% of the original price, since one tenth is 10%; a milk label saying 3.5% fat means every 100 g of milk holds 3.5 g of fat; a 70% chance of rain gives the likelihood of rain. Why learn to read them? Because percentages let us compare different ratio information: two juices labelled 30% and 45% fruit content are instantly ranked, the latter containing more fruit. A common mistake is to take 'one tenth off' as a flat 10 dollars off regardless of the price; it depends on the original price, and one tenth off 200 dollars is 20 dollars. When reading a percentage, always be clear about which total it refers to.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• A percentage is a fraction over 100, with the sign %
• 50% = 12 = 0.5; 25% = 14 = 0.25
• Percent to decimal: divide by 100 (dot two left)
• 100% is all; 0% is none; over 100% is possible
• Discounts, batteries and scores all use percentages