Percent (III): interest

1Summary 重點摘要

Interest = principal × rate × time.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Sort out the three wordsBefore computing interest, sort out three words: the principal is the money deposited, say $1000; the rate is the percentage earned each period, say 5% per year; the time is how long the money stays, say 2 years. Why sort them out? Because the three words occupy three positions in the formula, and mixing up any one puts a number in the wrong place and spoils the whole question; mistaking an annual 5% for a monthly rate inflates the interest twelve times. When practising, circle the principal, the rate and the time first, confirm them, and only then start computing; this is the fixed starting point of interest problems.
2. Step 2: The formulaSimple interest follows one line: interest = principal × rate × time. This formula links the three words together, and with it memorised every interest problem can be solved. Watch the units: the rate is per year, so the time must be measured in years too, and only then does the formula hold; for 5% a year over half a year, write 0.5 years, not 6 months. Why must the units agree? Because the rate is a per-year proportion, and the time in years is what matches it. A common mistake is mixing month and year, which inflates or shrinks the result twelvefold.
3. Step 3: One year's interestFind one year's interest first: 1000 × 5% = 1000 × 0.05 = $50. The rate 5% becomes the decimal 0.05, and multiplying it by the principal gives one year's interest. Why convert the percentage? Because the percent sign must be turned into a number before multiplying: 5% is 5100, that is 0.05. Many questions ask for one year's interest first, and this step starts most interest problems, with everything after built on it. A common error is skipping the conversion and computing 1000 × 5 = 5000, a hundred times too large; compare the sizes afterwards, since one year's interest can hardly exceed the principal itself.
4. Step 4: Multiply by the timeUnder simple interest every year earns the same, so 2 years earn 50 × 2 = $100. Whatever the number of years, multiply one year's interest by it. Why is every year the same? Because simple interest never adds the interest back into the principal; the base of the calculation stays at the original $1000, so each year brings in the same $50. This differs from compound interest, which grows year by year through interest on interest. A common mistake is assuming the interest increases annually and adding 50 + 100; under simple interest one multiplication by the number of years finishes the job.
5. Step 5: The total amountThe total amount is the principal plus the interest: 1000 + 100 = $1100. After two years the account grows from $1000 to $1100, and the extra $100 is the interest. Why distinguish interest from the total? Because word problems ask both ways, and answering the wrong object loses all the marks: if the question asks for the interest, the answer is only the extra $100; if it asks for the total, the answer is the final $1100. A common error is careless reading, giving the interest when the total was asked, or the reverse; circle what the question asks for before writing the answer, and attach the dollar unit.
6. Step 6: Another quick caseTry another example: principal $2000, annual rate 4%, kept for 1 year. One year's interest is 2000 × 0.04 = $80, and the total is $2080. With just one year the interest is simply the principal times the rate, the shortest form of the formula and the easiest calculation. Working several varied examples makes the use of principal, rate and time automatic, so the numbers fall into place on sight. A common error is a faulty percent-to-decimal conversion, such as writing 4% as 0.4 and getting $800, ten times too large; after converting, check the position of the decimal point before multiplying.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Interest = principal × rate × time
• Convert the rate to a decimal first (5% = 0.05)
• Total amount = principal + interest
• Simple interest reuses the original principal each year
• Check with rate × time as one total percent