Fractions

One of four prompts shows up: compare two fractions, simplify an unreduced one, or add/subtract a pair. Compare gets three giant buttons (<, =, >). The other three use the keypad — / types a fraction bar, Space separates a whole from a fraction in a mixed number. Answers must be in lowest terms: 6/8 gets rejected with a nudge to simplify to 3/4. A decimal equivalent (0.75) is accepted as an escape hatch. Press Enter to submit, ⌫ to undo, S or Space to skip.

Fractions are the one stretch of school math that tends to stick badly — and the damage lasts. A baker who rounds 3/8 cup to "about a third" ends up with flat cookies; a framer who treats 13/16 as "nearly an inch" cuts short. The fix isn't mysterious. Four moves — common denominators, simplification, cross-multiply comparisons, and mixed-number borrowing — cover roughly every fraction problem you'll encounter in a kitchen, a workshop, or a spreadsheet.

How do I add fractions with different denominators?

Short version: rewrite both fractions so they have the same bottom number — that shared bottom number is called a common denominator — then add the top numbers. The denominator (the bottom) tells you the size of each piece, and you can't add pieces of different sizes without restating them in the same size first.

How to find a common denominator. Any number that both denominators divide into evenly will work. The fastest way is to list multiples of the bigger denominator and stop at the first one the smaller denominator also divides evenly. For 4 and 6, multiples of 6 are 6, 12, 18, 24… — 12 is the first one 4 also divides evenly. That's the least common denominator, or LCD.

Shortcut if that's slow: just multiply the two denominators. 4 × 6 = 24 — both 4 and 6 divide into 24, so it's a valid common denominator. It's not the smallest, though, so the arithmetic is bigger and you'll have more simplifying to do at the end. Fine as a fallback; the LCD is tidier.

Walk through 3/4 + 5/6:

1. Find a common denominator. Multiples of 6: 6, 12, 18… — 4 divides evenly into 12. So the LCD is 12.
2. Rewrite each fraction with 12 on the bottom. For 3/4: we multiplied the bottom (4) by 3 to get 12, so multiply the top (3) by 3 too → 9/12. For 5/6: we multiplied the bottom (6) by 2 to get 12, so multiply the top (5) by 2 → 10/12. Whatever you do to the bottom you must do to the top — that's what keeps the fraction's value the same.
3. Add the top numbers. Bottoms stay the same: 9/12 + 10/12 = 19/12.
4. Convert to a mixed number if you like. 19 ÷ 12 = 1 remainder 7, so 19/12 = 1 7/12.

The step most people trip on is step 2 — forgetting to multiply the top by the same number you multiplied the bottom by. If you forget, the new fraction isn't equal to the old one anymore, and the answer comes out wrong. Whatever goes on the bottom, goes on the top. Every time.

What does simplifying a fraction mean?

Making the top and bottom numbers as small as they can be while keeping the fraction's value the same. You do this by dividing the top and bottom by the same number — whatever number divides evenly into both.

A quick vocabulary note: a factor of a number is anything that divides into it evenly. Factors of 6 are 1, 2, 3, 6. Factors of 8 are 1, 2, 4, 8. A common factor is one that shows up in both lists — 6 and 8 share 1 and 2.

Walk through simplifying 6/8:

1. List the factors. Top (6): 1, 2, 3, 6. Bottom (8): 1, 2, 4, 8.
2. Find the biggest one they share. Both have 2. Neither has anything bigger in common, so 2 is the greatest common factor (also called the greatest common divisor, or GCD).
3. Divide top and bottom by 2: 6 ÷ 2 = 3, 8 ÷ 2 = 4. Result: 3/4.
4. Check: do 3 and 4 still share a factor bigger than 1? No. Done — the fraction is in lowest terms.

If the GCD isn't obvious, chip away a factor at a time: divide by 2 while both are even, then by 3 while both digit-sums are divisible by 3, then by 5 while both end in 0 or 5. You'll land in the same spot, just in more steps.

Why do I have to find a common denominator?

Quick vocabulary: in the fraction 3/4, the 4 on the bottom is the denominator, and the 3 on the top is the numerator. The denominator tells you how many pieces a whole is split into; the numerator tells you how many of those pieces you have.

So the denominator is literally the size of each piece. In 1/2, the whole is split into two, so each piece is big. In 1/8, the whole is split into eight, so each piece is small. You can't add 1/2 to 1/3 directly as "two pieces" any more than you can add 50 cents to 33 cents as "83 coins" — the pieces aren't the same size.

The fix: restate both fractions so they're made of the same-sized pieces. That shared piece size is the common denominator. Once both fractions share it, the pieces line up and you can count them together.

The same logic applies on a ruler. You can't add 3/4" and 5/8" by just adding tops — the pieces aren't the same size. Restate both in sixteenths: 3/4" = 12/16", 5/8" = 10/16", sum 22/16" = 1 6/16" = 1 3/8". (This is exactly the move that saves lumber in our Tape Measure drill.)

How do I compare fractions quickly?

Cross-multiply. To compare 3/4 and 5/7, multiply across the diagonals:

1. 3 × 7 = 21 (top-left × bottom-right).
2. 5 × 4 = 20 (top-right × bottom-left).
3. The bigger product sits on the bigger fraction's side — 21 belongs to 3/4, so 3/4 > 5/7.

Why this works. You're quietly rewriting both fractions with a common denominator without showing the work. 3/4 becomes 21/28 (multiply top and bottom by 7); 5/7 becomes 20/28 (multiply top and bottom by 4). Now they share the same piece size — 28ths — and you just compare the tops: 21 vs 20. Cross-multiplying is a shortcut that skips writing the 28s out.

What's the fastest way to simplify a fraction?

Divide out obvious factors one at a time. Instead of trying to spot the biggest shared factor right away, chip away with small ones until nothing cancels. You'll land in the same place as finding the GCD, but the math is easier in your head.

The three divisibility tests do most of the work:

• Divides by 2 if the number is even (ends in 0, 2, 4, 6, or 8). 24 is even. 36 is even.
• Divides by 3 if the digit sum is divisible by 3. For 24: 2 + 4 = 6, which is divisible by 3. For 36: 3 + 6 = 9, also divisible by 3.
• Divides by 5 if the number ends in 0 or 5. 30 does; 36 doesn't.

Walk through 24/36:

1. Both even → divide top and bottom by 2 → 12/18.
2. Still both even → divide by 2 again → 6/9.
3. 6 and 9 — digit sums 6 and 9, both divisible by 3 → divide by 3 → 2/3.
4. 2 and 3 share no factor other than 1 → done. 24/36 = 2/3.

If after those three checks nothing cancels, the fraction is very likely already in lowest terms. (A prime numerator bigger than 5 is a dead giveaway — 7/12, 11/15, 13/24 are all already lowest terms.)

Why does 6/8 equal 3/4?

Because they describe the same share of the same whole — one just uses smaller pieces to say it. Imagine a chocolate bar. Snap it into 4 equal pieces and take 3 of them: that's 3/4. Now take the same bar, snap it into 8 equal pieces (half the size), and take 6 of them: that's 6/8. You're holding the exact same amount of chocolate either way.

Here's the rule that makes it work: if you multiply the top and the bottom of a fraction by the same number, the fraction's value doesn't change. 3/4 × 2/2 = 6/8. That looks like "multiplying by something" — but 2/2 is just 1 (two halves make a whole), and multiplying anything by 1 leaves it alone. So 3/4 and 6/8 are literally the same number.

The same logic gives you the whole family: 3/4 = 6/8 = 9/12 = 12/16 = 15/20… — every one of them is 3/4 multiplied by a different "sneaky 1" (3/3, 4/4, 5/5, and so on). All the same point on the number line, just different vocabulary.

How do I subtract mixed numbers when the fractional part is smaller?

Borrow one whole from the integer part — the same move you use to subtract two-digit numbers on paper.

Walk through 5 1/4 − 2 3/4:

1. 1/4 is smaller than 3/4, so you can't subtract the fractions directly.
2. Borrow a whole from 5: rewrite 5 1/4 as 4 + 1 + 1/4 = 4 + 4/4 + 1/4 = 4 5/4.
3. Now subtract: 4 5/4 − 2 3/4 = 2 2/4.
4. Simplify: 2 2/4 = 2 1/2.

If borrowing feels fussy, convert both numbers to improper fractions first: 5 1/4 = 21/4, 2 3/4 = 11/4, difference 10/4 = 2 1/2. Same answer, more arithmetic. Borrow in your head when the numbers are small; convert to improper when they're not.

When is an answer 'in lowest terms'?

When the numerator and denominator share no common factor except 1. 3/4 is in lowest terms (3 and 4 share no factors). 6/8 is not (both divisible by 2). If the numerator is a prime number that doesn't divide the denominator, the fraction is automatically in lowest terms — 7/12 is done the moment it lands. This drill won't advance on an unsimplified answer: it'll flag the fraction and let you try again.

Why bother with fractions when I can use decimals?

Because most real-world places you'd reach for a fraction can't be said cleanly as a decimal. 1/3 is 0.333… forever. 1/7 is uglier still. A tape measure reads 13/16, not 0.8125. A recipe calls for 3/4 cup, not 0.75 cup. Sheet music is in halves, quarters, eighths. Gears, wrench sets, drill bits, nails — everything sized in fractions of an inch. Decimals win for money and science; fractions win for anything physical you divide by cutting, folding, or counting.

Related drills on Practice

Tape Measure is where these fractions stop being abstract — sixteenths you can actually see. Percentages share the same "rewrite in a common unit" trick — a percentage is just a fraction with 100 always on the bottom. Quick Math keeps the multiplication fluency sharp; nothing slows a fractions problem down like fumbling 7 × 12. Making Change uses the same idea for coins: a fraction of a dollar, expressed in pennies. Memory Sprint drills the working-memory muscle you need when a fraction problem has more than one step.

As an Amazon Associate, we earn from qualifying purchases. We only link to things that are genuinely useful for the skill this page is about.

• Singapore Math fractions workbook The Singapore Math series teaches fractions with visual bar models before formulas — the single best fix for adults who've been doing fraction arithmetic on muscle memory and getting burned by the tricky ones.
• Fraction circle manipulatives Foam or plastic fraction circles — halves, thirds, quarters, sixths, eighths — for anybody who learns better by putting the pieces together than by reading about them. Worth it if you're teaching a kid; useful for adults too.
• Mental math tricks book Arthur Benjamin's "Secrets of Mental Math" has a tight chapter on fractions alongside the percent and multiplication chapters. Inexpensive, readable, the moves actually stick.
• Dry-erase lapboard set A small whiteboard for working problems by hand. Fraction arithmetic wants paper-and-pen space more than most skills — you'll stop dropping digits once you have somewhere to put them.