Fractions, decimals, and percents are three ways to describe parts of a whole. Students need this cheat sheet because the same value can look different, such as , , and . These skills are used in measurement, money, data, recipes, and word problems.
A clear reference helps students choose the right operation and convert between forms accurately.
The most important ideas are equivalent values, place value, and using common denominators. A percent means a number out of , so . Decimals use place values such as tenths, hundredths, and thousandths.
Fractions can be compared, added, subtracted, multiplied, and divided by following specific rules.
Key Facts
- A fraction means parts out of equal parts, where .
- Equivalent fractions are made by multiplying or dividing the numerator and denominator by the same nonzero number: .
- To add or subtract fractions with the same denominator, add or subtract only the numerators: .
- To add or subtract fractions with different denominators, first rewrite them with a common denominator, such as .
- To multiply fractions, multiply across: .
- To divide fractions, multiply by the reciprocal: , where and .
- To convert a percent to a fraction, use , and to convert a decimal to a percent, multiply by and add .
- To compare fractions, decimals, and percents, convert them to the same form, such as changing , , and to matching values.
Vocabulary
- Fraction
- A number in the form that shows part of a whole or part of a set.
- Numerator
- The top number in a fraction that tells how many parts are being counted.
- Denominator
- The bottom number in a fraction that tells how many equal parts make one whole.
- Decimal
- A number written with a decimal point to show parts based on powers of , such as tenths or hundredths.
- Percent
- A number that means parts per , written with the symbol .
- Equivalent Forms
- Different-looking numbers that have the same value, such as , , and .
Common Mistakes to Avoid
- Adding denominators, such as writing , is wrong because the denominator names the size of the parts and stays when the parts are the same size.
- Comparing fractions by only looking at numerators is wrong because is less than even though is greater than .
- Moving the decimal the wrong direction when converting to percent gives the wrong value because , not .
- Forgetting to use a common denominator before adding unlike fractions is wrong because cannot be added as .
- Dividing fractions without using the reciprocal is wrong because means how many halves fit in , so it becomes .
Practice Questions
- 1 Convert to a decimal and a percent.
- 2 Find and write the answer in simplest form.
- 3 Order , , and from least to greatest.
- 4 Explain why , , and represent the same amount.
Understanding Fractions, Decimals & Percents
A useful first step is to decide what the number is describing. In a recipe, a fraction may name part of a cup. On a receipt, a decimal usually names an amount of money.
In a sale, a percent tells how much of the original price changes. The form often gives a clue, but the size of the number still matters most. A discount of twenty five percent is one quarter of the price, while a decimal of zero point twenty five represents the same amount.
Estimation helps students notice mistakes before they become final answers. A value close to one half should not suddenly become a decimal near zero point zero five.
Place value controls the meaning of every decimal digit. The digit to the right of the decimal point represents tenths. The next digit represents hundredths.
Zeros matter because they hold positions in place. For example, zero point six is greater than zero point zero six because six tenths is greater than six hundredths. Adding zeros at the end does not change a decimal's value.
Zero point five, zero point five zero, and zero point five zero zero have equal value. This is helpful when comparing decimals.
Students can line up decimal points, then compare digits from left to right. Money gives a familiar example because dollars and cents use hundredths, even when an amount is written with only one cent digit.
Fraction operations make more sense when students connect them to units. Adding one third to one fourth cannot be done by adding the bottom numbers, because thirds and fourths are different sized pieces. Rewriting both amounts in equal sized pieces creates a shared unit for the calculation.
This is similar to adding lengths only after they use the same measurement unit. Multiplication has a different meaning. It often finds part of a part.
For instance, finding one half of three fourths means taking half of an amount that is already three fourths. The result must be smaller than three fourths.
Division can mean finding how many groups fit into an amount. Checking whether an answer is reasonable is especially important after division, since dividing by a number smaller than one can make the result larger.
Percent problems often involve finding a part, finding a whole, or finding the rate. A ten percent tip on a meal is easy to estimate because ten percent means one tenth. A fifteen percent tip can be found by combining ten percent with five percent.
In data displays, percent totals should usually be close to one hundred percent. Small differences can happen when values are rounded. When converting forms, students should keep the decimal point under control.
Moving from a decimal to a percent changes the scale from one whole to one hundred equal parts. A percent greater than one hundred is possible.
It means more than the original whole, such as a score that increased by one hundred twenty percent. Students should write each conversion step clearly and use a benchmark such as zero, one half, one, fifty percent, or one hundred percent to check the result.