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Hong KongFractions, decimals and percents
The rule folds these into "the four operations with rational numbers", so they are one skill rather than a topic.
- They turn up everywhere else too: ratio problems, data interpretation and measurement all use them.
- Conversion in both directions is the technique worth automating.
On this page9 sections
- The conversion triangle
- The set worth knowing without the calculator
- The three percent questions
- Percent change, and why it is not symmetric
- Fraction operations, briefly
- Where the calculator helps and where it does not
- Convert among equivalent forms
- Compare values using a common form
- Calculate a percent of a quantity
One skill in the rule, and the arithmetic behind a large share of the paper. That gap is why this page exists.
The conversion triangle
| From | To | How |
|---|---|---|
| Fraction | Decimal | Divide top by bottom |
| Decimal | Percent | Multiply by 100 |
| Percent | Decimal | Divide by 100 |
| Decimal | Fraction | Write over 10 or 100, then reduce |
| Fraction | Percent | Divide, then multiply by 100 |
With a calculator, every one of these is a keystroke. What it will not do is tell you which conversion you need, and that is where the marks go.
The set worth knowing without the calculator
- A half is 50 percent
- A fifth is 20 percent
- A tenth is 10 percent
- Three fifths is 60 percent
- Seven tenths is 70 percent
Knowing these on sight is not about speed. It is about sanity-checking an answer the calculator produced from a keystroke you mistyped.
The three percent questions
| Question | Setup | Example |
|---|---|---|
| What is x percent of y? | Multiply | 20 percent of 45 is 9 |
| X is what percent of y? | Divide, then times 100 | 9 out of 45 is 20 percent |
| X is y percent of what? | Divide by the decimal | 9 is 20 percent of 45 |
All three are the same equation rearranged: part equals percent times whole. Write that down at the start of a percent question and you will never pick the wrong operation.
A class of 20 students has 12 who ride the bus. What percentage ride the bus?
- 12 percent
- 20 percent
- 40 percent
- 60 percent
Percent change, and why it is not symmetric
Percent change is the difference divided by the original amount. The original amount is the part people get wrong.
Go from 50 to 100 and that is a rise of 50 on a base of 50, which is 100 percent. Go from 100 back to 50 and that is a fall of 50 on a base of 100, which is 50 percent. Same two numbers, different percentages, and an item can be built on exactly that asymmetry.
Fraction operations, briefly
Add and subtract with a common denominator. Multiply straight across. Divide by inverting the second fraction and multiplying.
The one to keep straight is division, because it is the only operation where you change one of the numbers before you start. Two thirds divided by a half is two thirds times two, which is four thirds. The answer got bigger, which is correct and looks wrong.
Where the calculator helps and where it does not
The Test Structure document says an on-screen calculator is supplied for this subtest. It will do the arithmetic perfectly and it will happily compute the wrong thing, so the constraint on this competency is setup, not calculation.
With 2.86 minutes a question on our division of the published figures, you have time to write the relationship down before you touch it. Most people do not, and that is where the marks go on a subtest that 71% of first-time candidates pass.
Convert among equivalent forms
A percent means per hundred, so 37% = 37/100 = 0.37. To convert a terminating decimal to a percent, multiply by 100 and add the percent sign: 0.08 = 8%. To convert a percent to a decimal, divide by 100: 4.5% = 0.045. Keep track of the decimal point; moving it in the wrong direction changes the value by a factor of 100.
To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a fraction, use place value and reduce: 0.375 = 375/1000 = 3/8. A fraction can be converted to a percent by multiplying by 100: 3/8 × 100% = 37.5%. You can also find an equivalent fraction with denominator 100 when convenient.
Compare values using a common form
To compare 2/3, 0.65, and 68%, convert them to decimals: about 0.667, 0.65, and 0.68. Thus 68% is greatest. Exact fractions can be compared by using a common denominator or cross-multiplication; with positive denominators, compare the products of opposite numerator and denominator. For 3/5 and 5/8, compare 3 × 8 = 24 with 5 × 5 = 25, so 5/8 is larger.
Calculate a percent of a quantity
“What is 18% of 250?” means multiply 0.18 × 250 = 45. “45 is what percent of 250?” means divide part by whole: 45/250 = 0.18 = 18%. “45 is 18% of what number?” means solve 0.18x = 45, so x = 250. Identify which quantity is the part, the whole, and the rate before choosing multiplication or division.
For a discount or tax, convert the percent to a decimal and apply it to the correct base. A 20% discount on $80 is $16, leaving $64. A 6% tax on a $64 sale is $3.84, so the total is $67.84. If the problem asks for a percent increase or decrease, compare the change with the original amount, not the final amount.
- Percent means per hundred; divide by 100 to convert it to a decimal.
- Convert values to a common form before comparing.
- Identify the part, whole, and percent in word problems.
- Use a decimal multiplier for percent-of calculations.
- For percent change, divide the change by the original value.
Common questions
How do I convert a fraction to a percentage?
Divide the top by the bottom to get a decimal, then multiply by 100. Three fifths comes out as 60 percent. With a calculator that is two keystrokes, so the difficulty is deciding which number goes on top rather than doing the arithmetic itself.
What is the formula for percent change?
The difference divided by the original amount. Going from 50 to 100 is a rise of 50 on a base of 50, or 100 percent. Going from 100 back to 50 is a fall of 50 on a base of 100, or 50 percent. The base changes, so the percentage does.
Do I need to know fraction and decimal equivalents by heart?
Not to compute them, since a calculator is supplied. Knowing the common ones on sight is worth it as a check, because a mistyped keystroke produces a confident wrong answer and only familiarity with the expected size will catch it.
Are fractions their own topic on the FTCE?
No. The rule includes them under solving real-world problems with the four operations on rational numbers, which is one skill of twenty-one. They then reappear inside the ratio, measurement and data competencies, which is why they are worth practicing anyway.