Packets  /  Fractions Master
Grades 3-5Elementary

Fractions Master

Master fractions from the ground up: what they mean, equivalent fractions, simplifying, comparing, adding, subtracting, multiplying, dividing, mixed numbers, and real-world word problems. Learn each idea, practice with instant feedback, then beat the 60-second challenge.

1What a Fraction Means

Key idea: A fraction (a/b) names part of a whole: the bottom (denominator) counts how many EQUAL parts the whole is split into, and the top (numerator) counts how many of those parts you have.
  1. Split one whole into equal parts — they must all be the same size.
  2. The denominator (bottom number) is the total number of equal parts.
  3. The numerator (top number) is how many of those parts you are counting.
  4. Read the fraction top first, then bottom: 34 is 'three fourths'.
Worked example
34 of a pizza
The pizza is cut into 4 equal slices, and 3 slices are eaten. Top (numerator) 3 = parts we count; bottom (denominator) 4 = total equal parts.
Worked example
12 of an hour
The hour is split into 2 equal parts, and we count 1 of them. That is one half: 30 minutes.
Watch out: The parts MUST be equal! If a pizza is cut into 4 uneven pieces, eating 1 piece is NOT (1/4). Only equal pieces count as fractions.

2Equivalent Fractions

Key idea: Multiplying or dividing the top AND bottom by the SAME number gives an equivalent fraction — a different name for the exact same amount.
  1. Start with your fraction, for example 12.
  2. Pick any counting number (2, 3, 4, …).
  3. Multiply the top and the bottom by that number.
  4. The new fraction names the same amount: 12 = 24 = 36 = 48.
Worked example
12 = (?) / (4)
Multiply top and bottom by 2: (1 x 2) / (2 x 2) = 24. Same amount, different name.
Worked example
23 = (?) / (12)
Multiply top and bottom by 4: (2 x 4) / (3 x 4) = 812. Check: both name the same amount.
Watch out: You must multiply or divide BOTH the top and the bottom! Multiplying only the top changes the value — (1/2) is not equal to (2/2).

3Simplifying to Lowest Terms

Key idea: To simplify, divide the top and bottom by their greatest common factor (GCD) until no number except 1 divides both. That is lowest terms.
  1. Find the GCD — the biggest number that divides both the top and the bottom.
  2. Divide the top by the GCD.
  3. Divide the bottom by the GCD.
  4. Check: if any number except 1 still divides both, keep going.
Worked example
Simplify 48
The GCD of 4 and 8 is 4. Divide top and bottom by 4: 44 / 84 = 12. Simplest form!
Worked example
Simplify 1216
The GCD of 12 and 16 is 4. Divide: 124 / 164 = 34. No number except 1 divides both 3 and 4, so we are done.
Watch out: (2/4) is NOT finished — both 2 and 4 are still divisible by 2, giving (1/2). A fraction is only simple when nothing except 1 divides both numbers.

4Comparing and Ordering Fractions

Key idea: Same bottom: the bigger top wins. Same top: the SMALLER bottom wins (bigger slices). Otherwise, convert to a common bottom first, then compare tops.
  1. Check the bottoms: if they match, circle the fraction with the bigger top.
  2. Check the tops: if they match, circle the fraction with the smaller bottom.
  3. If neither matches, rewrite both with a common bottom.
  4. Then compare the tops to decide which is bigger.
Worked example
Which is bigger: 38 or 58?
Same bottom! Bigger top wins: 58 is bigger than 38.
Worked example
Which is bigger: 23 or 25?
Same top! Smaller bottom wins because the slices are bigger: 23 is bigger than 25.
Worked example
Order 14, 12, 34 from smallest to biggest
Convert to the same bottom: 14, 24, 34. Smallest to biggest: 14, 12, 34.
Watch out: Same top does NOT mean same size! (2/5) is smaller than (2/3) because fifths are tinier slices than thirds. The bigger bottom makes SMALLER pieces.

5Adding and Subtracting Fractions

Key idea: Same bottom: just add or subtract the TOPS — the bottom never changes. Different bottoms: convert both to a common bottom first, then add or subtract the tops.
  1. Look at the bottoms. If they match, add or subtract the tops and keep the bottom.
  2. If they do not match, find a common bottom (the LCD works best).
  3. Rewrite each fraction with the common bottom.
  4. Add or subtract the tops, keep the bottom, then simplify.
Worked example
27 + 37
Same bottom! Add the tops: (2 + 3) / 7 = 57. The bottom stays 7.
Worked example
79 – 29
Same bottom! Subtract the tops: (7 – 2) / 9 = 59. The bottom stays 9.
Worked example
12 + 14
Different bottoms! Convert: 12 = 24. Then add the tops: 24 + 14 = 34.
Worked example
34 – 12
Convert 12 = 24. Then subtract the tops: 34 – 24 = 14.
Watch out: NEVER add or subtract the bottoms! (1/4) + (1/4) = (2/4), NOT (2/8). The bottom tells the slice size — adding slices does not change the slice size.

6Multiplying Fractions

Key idea: Multiply straight across: top times top over bottom times bottom. For a whole number, write it over 1 first.
  1. Write any whole number as a fraction over 1 (3 becomes 31).
  2. Multiply the tops together to get the new top.
  3. Multiply the bottoms together to get the new bottom.
  4. Simplify the answer. Tip: cancel common factors BEFORE multiplying to keep numbers small.
Worked example
12 x 14
Multiply straight across: (1 x 1) / (2 x 4) = 18.
Worked example
3 x 25
Write 3 as 31. Multiply across: (3 x 2) / (1 x 5) = 65.
Worked example
23 x 34
Across: (2 x 3) / (3 x 4) = 612. Simplify: 12. Faster: cancel the 3s first, then 24 = 12.
Watch out: Do NOT find a common denominator for multiplication — that rule is only for adding and subtracting. Multiply straight across.

7Dividing Fractions

Key idea: To divide fractions: Keep the first fraction, Change division to multiplication, Flip the second fraction (use its reciprocal). Then multiply.
  1. Keep the first fraction exactly as it is.
  2. Change the division sign to multiplication.
  3. Flip the second fraction — swap its top and bottom.
  4. Multiply straight across, then simplify.
Worked example
12 / 14
Keep-Change-Flip: 12 x 41 = 42 = 2. Half a pizza holds two quarter slices!
Worked example
34 / 12
Keep-Change-Flip: 34 x 21 = 64. Simplify: 32.
Watch out: Flip ONLY the second fraction, never the first! (2/3) / (1/2) becomes (2/3) x (2/1), not (3/2) x (1/2).

8Mixed Numbers and Improper Fractions

Key idea: Improper fractions have a top bigger than the bottom (more than one whole). Mixed numbers pair a whole number with a fraction. Convert: improper to mixed by dividing (the remainder is the new top); mixed to improper with whole x bottom + top, all over the same bottom.
  1. Improper to mixed: divide the top by the bottom. The quotient is the whole number, the remainder is the new top, the bottom stays.
  2. Mixed to improper: multiply the whole number by the bottom, then add the top — that is the new top.
  3. Keep the same bottom in both directions.
  4. Simplify the fraction part if you can.
Worked example
Write 74 as a mixed number
Divide: 7 / 4 = 1 with remainder 3. So 74 = 134. The bottom stays 4.
Worked example
Write 213 as an improper fraction
Whole x bottom, then add the top: 2 x 3 + 1 = 7. So 213 = 73. The bottom stays 3.
Watch out: The bottom NEVER changes when converting! (7/4) = 1(3/4), not 1(3/7) or 1(7/3).

9Solving Fraction Word Problems

Key idea: Read the story twice, underline what you must find, and pick the operation from the action: join means add, take away means subtract, groups of means multiply, share equally means divide. Then solve, simplify, and check the units.
  1. Read the problem twice and underline the question being asked.
  2. Spot the action word: 'in all' or 'join' means add; 'left' or 'take away' means subtract; 'groups of' or 'times' means multiply; 'share equally' means divide.
  3. Write the fraction number sentence and solve it.
  4. Simplify your answer and include the units.
  5. Check: does the answer make sense in the story?
Worked example
Ana drank 12 liter of juice and Sam drank 14 liter. How much did they drink in all?
'In all' means join, so add: 12 + 14. Convert: 24 + 14 = 34. They drank 34 liter in all.
Worked example
3 friends share 34 of a pizza equally. How much does each friend get?
'Share equally' means divide: 34 / 3. Keep-Change-Flip: 34 x 13 = 312 = 14. Each friend gets 14 of the pizza.
Watch out: Do not rush to add just because you see two fractions! The action decides the operation — 'share equally' means divide even when addition feels tempting.
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60-Second Challenge

How many fractions master problems can you solve in 60 seconds?