Equivalent Fractions, Mixed Numbers and Comparing Fractions

Imagine folding a paper strip in half and shading one half. Fold it again, then open it. You now see four equal parts, with two shaded. The shaded amount stays the same.

Match the shaded lengths

Equivalent fractions name the same number. For example, $\frac{1}{2}=\frac{2}{4}=\frac{4}{8}$.

Make three strips of the same length. Split them into halves, fourths, and eighths. Shade one half, two fourths, and four eighths. Line up the left ends. All three shaded lengths end at the same place.

The eighths strip has more shaded pieces than the halves strip. Each eighth is much smaller than a half. Counting pieces alone will not tell you how much is shaded.

Split a shaded shape

Draw a rectangle with four equal columns. Shade three columns to show $\frac{3}{4}$. Draw a line across the middle of the rectangle. Each fourth becomes two equal pieces.

Now there are eight equal pieces in the whole rectangle. Six are shaded, so the picture also shows $\frac{6}{8}$. The new lines let you count the same shaded area in smaller pieces.

Make every part equal in size. Four uneven pieces do not show fourths.

Find one point with two names

Mark 0 and 1 on a number line. Split that distance into three equal spaces. The second mark is $\frac{2}{3}$.

Split each space in half. The line now has six equal spaces. The same point is four spaces from 0, so it is also $\frac{4}{6}$. Equivalent fractions share a point; you do not move the point to rename it.

A fraction family collects equal fractions, such as $\frac{1}{3}=\frac{2}{6}=\frac{3}{9}=\frac{4}{12}$. On a fraction wall, their shaded ends line up. One third does not line up with any whole number of fourths.

Keep the whole the same size

Half of a large sandwich can be more food than half of a small sandwich. The number one half stays the same, but the sandwiches have different sizes. To compare shaded amounts, use matching wholes.

One whole has fraction names too: $\frac{2}{2}=\frac{4}{4}=\frac{8}{8}=1$. You have all the parts when the numerator matches the denominator.

You can give a fraction a new name without drawing every piece. Multiply its numerator and denominator by the same whole number greater than zero.

Count smaller pieces

Start with $\frac{2}{3}$. Cut every third into four equal pieces. The whole now has twelve pieces. Each of the two shaded thirds holds four pieces, so eight are shaded.

Multiplying the denominator tells how many smaller parts fill the whole. Multiplying the numerator counts how many of those parts you have. Both numbers must change together.

You can also think of $\frac{4}{4}$ as one whole. Multiplying a number by one keeps its value unchanged.

Fill a missing numerator

Find the missing number in $\frac{3}{4}=\frac{?}{20}$.

1. Look at the denominators: $4\times 5=20$.
2. Use the same multiplier on top: $3\times 5=15$.
3. Write $\frac{3}{4}=\frac{15}{20}$.

Adding 16 to the bottom does not mean adding 16 to the top. The rule uses multiplication, not matching amounts added.

Fill a missing denominator

For $\frac{5}{6}=\frac{25}{?}$, start with the numerators. Five becomes twenty-five by multiplying by five. Multiply the denominator by five too: $6\times 5=30$. The missing denominator is 30.

In $\frac{3}{4}=\frac{9}{12}$, both numbers were multiplied by three. Changing the denominator to sixteen would break the match.

Build a family

Multiply both numbers in $\frac{3}{5}$ by two, three, then four:

For tenths to hundredths, the multiplier is ten. That gives $\frac{7}{10}=\frac{70}{100}$. Useful family members include $\frac{1}{2}=\frac{50}{100}$, $\frac{1}{4}=\frac{25}{100}$, and $\frac{1}{5}=\frac{20}{100}$.

Check a changed fraction

Doubling only the denominator of $\frac{2}{3}$ gives $\frac{2}{6}$. You still have two pieces, but each is half as big. The amount has changed. To keep the amount equal, double both numbers and write $\frac{4}{6}$.

Six eighths and three fourths cover the same amount. The second name uses smaller numbers. Simplifying means finding an equal fraction this way.

Divide both numbers

Divide the numerator and denominator by the same common factor. A factor divides a number exactly, with nothing left over.

For $\frac{6}{8}$, both numbers divide by two:

Picture grouping the eighths in pairs. Two eighths make one fourth. Six eighths make three fourths, and the whole holds four fourths.

Know when to stop

A fraction is in simplest form when its only common factor is one. Three and four share no greater factor, so $\frac{3}{4}$ is finished. Simplest form is also called lowest terms.

Try $\frac{12}{18}$. Both numbers are even, so divide by two:

Six and nine both divide by three. Keep going:

Two and three share only one as a factor. Stopping at six ninths would give the right amount, but would leave more simplifying to do.

Take a shortcut with factors

The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. Their shared factors are 1, 2, 3, and 6.

Six is the greatest common factor: the largest shared factor. Dividing twelve and eighteen by six reaches $\frac{2}{3}$ in one step. Smaller steps work just as well when you check again at the end.

Spot helpful patterns

For $\frac{20}{100}$, divide both numbers by ten, then by two:

If the numerator divides the denominator exactly, you can reach a unit fraction. For example, $\frac{5}{15}=\frac{1}{3}$ after dividing by five. Neighboring counting numbers, such as seven and eight, share no factor above one. So $\frac{7}{8}$ is already in simplest form.

Simplifying can also reveal equal fractions. Four sixths and six ninths both simplify to two thirds.

Subtracting the same number from top and bottom is different. Taking six from both numbers in $\frac{12}{18}$ gives $\frac{6}{12}$, or one half. That changes the amount. Simplifying uses division.

A fraction can name less than a whole, exactly a whole, or several wholes. The denominator still tells the size of the parts. The numerator counts how many you have.

Three ways to write amounts

Read $2\frac{3}{4}$ as “two and three-fourths.” You can also say “two and three quarters.” The space joins a whole amount and extra parts.

Build more than one whole

Picture pizzas of the same size, each cut into four equal slices. Seven slices give seven fourths. Four slices make one pizza, leaving three slices.

The word improper names a kind of fraction. It does not mean the fraction is wrong.

Count along a number line

Split every whole-number space into four equal spaces. Count fourths from zero:

You reach one at four fourths and two at eight fourths. Five fourths is one step past one, so it equals $1\frac{1}{4}$. Keep the spacing equal after passing one.

To place $2\frac{1}{3}$, find two first. Split the space from two to three into three equal spaces. Move one space right from two. The point is between two and three, closer to two. It also has the name $\frac{7}{3}$.

Recognize complete wholes

An improper fraction can be a whole number. For $\frac{12}{4}$, twelve fourths fill three wholes. Any whole number also has a fraction name with denominator one: $3=\frac{3}{1}$.

You can simplify a mixed number's fraction part: $1\frac{2}{4}=1\frac{1}{2}$. If that part holds another whole, take it out. One whole plus five fourths is two wholes plus one fourth, written $2\frac{1}{4}$.

An improper fraction counts all the pieces together. To write a mixed number, find how many full wholes those pieces can fill.

Divide to find the groups

Take $\frac{17}{5}$. Each whole needs five fifths. Ask how many groups of five fit into seventeen.

Three groups use fifteen fifths, leaving two fifths. So:

The quotient, three, gives the full wholes. The remainder, two, counts leftover pieces. Those pieces are still fifths, so the denominator stays five. The fraction bar itself can mean division: seventeen fifths is seventeen divided by five.

Use a times-table fact

For $\frac{23}{4}$, look for the largest multiple of four below twenty-three.

1. $4\times 5=20$, so five wholes use twenty fourths.
2. $23-20=3$, so three fourths remain.
3. Write $\frac{23}{4}=5\frac{3}{4}$.

The remainder is smaller than the denominator. Four leftover fourths would make another whole, so they could not stay in the fraction part.

Finish the fraction part

For $\frac{14}{4}$, division gives three wholes and two fourths. Simplify the two fourths to one half:

You could simplify first instead: fourteen fourths becomes seven halves. Dividing seven by two also gives three and one half.

When the division leaves nothing over, write a whole number. For $\frac{18}{6}$, eighteen divided by six is three exactly. The answer is three.

Check by counting the parts again

For $5\frac{3}{4}$, the five wholes hold twenty fourths. Add the three extra fourths to get twenty-three fourths. This returns to the fraction you started with.

A mixed number often makes a measurement easier to use. Nine fourths of a cup is $2\frac{1}{4}$ cups: two full cups and one fourth of a cup. Keep the measurement unit beside the answer.

Watch where the leftover number goes. It counts pieces, so it belongs on top of the fraction. The denominator names the pieces and stays unchanged.

A mixed number separates the wholes from the extra pieces. To write one improper fraction, count the pieces inside the wholes too.

Open up the wholes

Start with $2\frac{3}{4}$. Each whole holds four fourths. Two wholes therefore hold eight fourths. Add the three fourths beside them:

There are eleven fourths altogether, so $2\frac{3}{4}=\frac{11}{4}$.

You could draw two whole circles split into fourths and three extra fourths. Counting all the fourths would give the same eleven pieces.

Multiply, then add

For any mixed number:

1. Multiply the whole number by the denominator.
2. Add the numerator to count the extra pieces.
3. Put the total over the same denominator.

For $3\frac{2}{5}$, the three wholes hold $3\times 5=15$ fifths. Two more fifths make seventeen. Write $\frac{17}{5}$.

The denominator stays five because you are still counting fifths. Only the way you group the pieces has changed.

Keep track of what each number means

For $4\frac{1}{6}$, four is the number of wholes. Six tells how many sixths fill each whole. One counts the extra sixth.

Adding four and one would count five, but would skip the sixths hidden inside the wholes. Multiplying the whole number by the numerator also misses those pieces. Start with the denominator: it tells how many pieces each whole holds.

Check the size

Twenty-five sixths should be between four and five. Four wholes use twenty-four sixths, leaving one sixth. Dividing twenty-five by six brings you back to $4\frac{1}{6}$.

Changing form can help you compare two amounts. For $2\frac{1}{2}$ and $\frac{9}{4}$, first write $2\frac{1}{2}=\frac{5}{2}$. Rename five halves as ten fourths. Ten fourths is more than nine fourths, so $2\frac{1}{2}>\frac{9}{4}$.

When two fractions have something in common, you can often compare them quickly. First decide whether their denominators or their numerators match.

Same denominator: count the pieces

For $\frac{5}{8}$ and $\frac{3}{8}$, every piece is an eighth. Five equal pieces make more than three of those pieces:

You can use the same idea above one whole. Nine fourths is more than seven fourths, so $\frac{9}{4}>\frac{7}{4}$. Crossing a whole-number mark does not change the size of a fourth.

Same numerator: compare piece sizes

For $\frac{3}{4}$ and $\frac{3}{8}$, each amount has three pieces. Fourths are bigger than eighths, so three fourths is greater.

Draw two equal-length strips to see why. Cut one into four equal parts and the other into eight. Shade three parts on each. The strip with fourths has a longer shaded length.

A bigger denominator means the whole was cut into more equal pieces. Each piece is smaller. For example, $\frac{1}{8}<\frac{1}{4}$, even though eight is greater than four.

Read comparison signs

The sign $>$ means “is greater than.” The sign $<$ means “is less than.” The open side faces the greater number; the pointed end faces the smaller number.

Both $\frac{3}{5}>\frac{2}{5}$ and $\frac{2}{5}<\frac{3}{5}$ compare the same two amounts correctly. Use $=$ for equivalent fractions, such as $\frac{2}{4}=\frac{1}{2}$.

Compare matching wholes

As numbers, one half is less than three fourths. To compare actual pieces of food with those fractions, the starting wholes must be the same size. Half of a large pizza could be more food than three fourths of a small pizza. Matching fraction strips avoid this problem because their whole lengths agree.

You do not always need to rename fractions to compare them. A familiar number can help you see which amount is greater. Such a number is called a benchmark.

Find each side of one half

Half of eight is four. So four eighths is one half. Three eighths is below one half, and five eighths is above it.

For odd denominators, doubling the numerator is handy. To check $\frac{4}{7}$, double four to get eight. Eight is greater than seven, so four sevenths is more than half.

This gives a quick comparison:

Three eighths is on the smaller side of the same marker. Four sevenths is on the greater side. Therefore, three eighths is less than four sevenths.

Doubling also spots exact halves. For three sixths, twice three equals six, so $\frac{3}{6}=\frac{1}{2}$.

Check one whole

A fraction with a numerator below its denominator is below one. A greater numerator puts it above one. This settles $\frac{6}{5}>\frac{9}{10}$: six fifths is above one, while nine tenths is below it.

Look at the missing piece

Five sixths is one sixth short of a whole. Seven eighths is one eighth short. An eighth is smaller than a sixth. Seven eighths has less left to reach one, so it is greater:

Being “one piece short” does not make fractions equal. A missing third is much larger than a missing sixth. The size of that missing piece matters.

Notice when you need another method

Three fifths and two thirds are both above one half. Knowing that does not tell you their order. You need a closer comparison.

In fifteenths, $\frac{3}{5}=\frac{9}{15}$ and $\frac{2}{3}=\frac{10}{15}$. Now the pieces match, and $\frac{3}{5}<\frac{2}{3}$.

Choose a benchmark when it puts the fractions on opposite sides. If both are on the same side, try equal fraction names or compare their distances from one.

Two fractions may have different numerators and different denominators. Give them equivalent names with matching denominators. Then their pieces will be the same size.

Change one fraction when you can

Compare $\frac{3}{4}$ and $\frac{5}{8}$. Four goes into eight, so use eighths.

Six eighths is more than five eighths. That means $\frac{3}{4}>\frac{5}{8}$.

Multiply both numbers when renaming. Changing three fourths to three eighths would shrink its value and spoil the comparison.

Find a shared denominator

A common denominator is a shared multiple of the denominators. For thirds and fourths, list multiples until you find a match:

• Multiples of three: 3, 6, 9, 12.
• Multiples of four: 4, 8, 12.

Use twelve to compare $\frac{2}{3}$ and $\frac{3}{4}$:

Eight twelfths is less than nine twelfths. Write the result using the original fractions: $\frac{2}{3}<\frac{3}{4}$.

Keep the numbers small

Multiplying the denominators always gives a common denominator. For sixths and fourths, twenty-four works. Twelve works too and keeps the numbers smaller.

To compare five sixths with three fourths, use $\frac{5}{6}=\frac{10}{12}$ and $\frac{3}{4}=\frac{9}{12}$. Five sixths is greater. The larger shared denominator would give the same order.

Sometimes the result is equality. Six ninths and four sixths both become twelve eighteenths, so they are equal. Simplifying both to two thirds shows the same match.

Or match the numerators

For $\frac{2}{5}$ and $\frac{4}{9}$, double both numbers in two fifths. You get four tenths.

Now compare four tenths with four ninths. Each has four pieces, but ninths are bigger. So $\frac{2}{5}<\frac{4}{9}$.

Matching numerators works by comparing piece size. Matching denominators works by comparing piece count. In either method, the renamed fractions must stay equal to the originals.

Ordering several fractions uses the same comparisons as ordering two. Work in small groups, and keep track of the direction you need.

Least to greatest or greatest to least

Ascending order runs from least to greatest. Descending order runs from greatest to least. On a number line, values grow as you move right.

For $\frac{1}{2},\frac{3}{8},\frac{3}{4},\frac{5}{8}$, write the values in eighths. They become four, three, six, and five eighths. Sort those counts:

Write the original names in the answer. The equal names were a way to compare them.

Sort by size first

Benchmarks can reduce the work. Put fractions below one half together. Make another group between one half and one, and another above one. Compare closely within each group.

Mixed numbers show their whole parts straight away. Three and one fifth is greater than two and seven eighths. Any amount between three and four is greater than an amount between two and three.

If the whole parts match, compare the fractions. For $2\frac{3}{4}$ and $2\frac{2}{3}$, compare nine twelfths with eight twelfths. The first mixed number is greater.

Handle a list with different forms

Order $\frac{11}{4},2\frac{1}{2},2\frac{3}{8},\frac{9}{4}$ from least to greatest.

1. Change the improper fractions: $\frac{11}{4}=2\frac{3}{4}$ and $\frac{9}{4}=2\frac{1}{4}$.
2. All four numbers have two wholes. Compare their fraction parts in eighths: six, four, three, and two eighths.
3. Put the smallest fraction part first:

On a number line, split the space from two to three into eight equal spaces. These numbers sit two, three, four, and six spaces past two.

Check neighboring numbers

Read your ordered list one pair at a time. Each value must grow in ascending order. For descending order, each value must get smaller.

A mixed number is not always greater than an improper fraction. Compare their amounts, not how they look. Equal fractions share a position: place $\frac{3}{6}$ and $\frac{1}{2}$ together with an equals sign.

Three groups of two fifths contain six fifths altogether. Multiplication counts repeated fraction amounts, just as it counts repeated whole-number amounts.

Count unit fractions

A unit fraction has one on top. Five fourths means five copies of one fourth:

For three groups of two fifths, each group contains two one-fifth pieces. Count two, four, six fifths:

Keep the part size

Multiply the numerator by the whole number. Keep the denominator because fifths stay fifths.

Twelve eighths fill one whole and leave four eighths. Simplify the fraction part:

Multiplying both numerator and denominator would give an equivalent name for the starting fraction. It would not find four groups of that amount. Here you need more eighths, not smaller pieces.

Make equal jumps

On a number line, start at zero. For $3\times \frac{2}{5}$, make three jumps of two fifths. Land first at two fifths, then four fifths, then six fifths. The final point is $1\frac{1}{5}$.

You can reverse the multiplication order: $\frac{2}{5}\times 3$ gives the same result. For these positive fraction amounts, taking more than one group gives more than a single group.

Multiply a mixed amount

For $3\times 1\frac{1}{4}$, change the mixed number into five fourths. Multiply to find fifteen fourths, then group them into wholes:

You can also count the whole and fraction parts separately. Three groups each have one whole and one fourth. That makes three wholes and three fourths.

Keep both parts of the mixed number in your calculation. Multiplying just the fourth would leave out all three wholes.

Look for an amount repeated the same number of times. A story about several equal servings or lengths often calls for multiplication.

Find the two amounts you need

Five friends each drink three fourths of a cup of juice. The group count is five. The amount in each group is three fourths of a cup.

Fifteen divided by four gives three with three left over. The friends drink $3\frac{3}{4}$ cups altogether. Adding five to three fourths would count five full cups plus one serving, which is a different amount.

A tape diagram can help. Draw five bars, each showing three fourths of a cup. Together they contain fifteen fourth-cup pieces. Four of those pieces fill each cup.

Estimate the total

Each friend drinks less than a full cup. Five friends therefore drink less than five cups. Each serving is also more than half a cup, so the total is more than two and one-half cups. Three and three-fourths cups fits those limits.

To find which neighboring whole numbers surround the total, read the mixed number. Three and three-fourths cups is between three and four cups.

Some totals are exact wholes. Six walks of two thirds of a mile total $\frac{12}{3}=4$ miles. Say four miles exactly; there is no leftover fraction.

Work with mixed servings

A recipe uses one and one-half cups of flour. Making it three times needs three equal groups of that amount.

You need four and one-half cups of flour. The unit stays cups throughout.

Calculate before comparing

Ana runs three fourths of a mile three times. Ben runs two and one-half miles. Ana's total is $3\times \frac{3}{4}=\frac{9}{4}=2\frac{1}{4}$ miles.

Both totals have two wholes. One fourth is less than one half, so Ben runs farther. Comparing one of Ana's runs with Ben's full distance would leave out two of her runs.

Before writing your answer, check what the story asks for: a total, a comparison, or the whole numbers around a total. Include the unit so your answer names the actual amount.