Fractions

Study Sheet

Fractions

Everything you need to master parts of a whole

Fraction Vocabulary & Equivalent Fractions

Concept
The Parts of a Fraction
2/34/66/9
Thirds, sixths, ninths: the shaded amount is the same, 23=46=69\tfrac23 = \tfrac46 = \tfrac69.

A fraction names part of a whole (or part of a group). It is written as numeratordenominator\dfrac{\text{numerator}}{\text{denominator}}.

  • The denominator (bottom) tells how many equal pieces the whole is divided into.
  • The numerator (top) tells how many of those pieces you have.

In 34\dfrac{3}{4}, the whole is cut into 44 equal parts and we have 33 of them. Two fractions are equivalent when they name the same amount, even though the numbers look different. You make an equivalent fraction by multiplying (or dividing) the top and bottom by the same nonzero number.

Fractions add and subtract only over a common denominator, multiply straight across, and divide by multiplying by the reciprocal. Mixed numbers are improper fractions in costume; converting first avoids most errors. Comparing fractions is easiest with a common denominator or by cross-multiplying.

Equivalent fractions are the same amount cut into different numbers of pieces. Cutting each third of a bar in half doubles the number of pieces (3→63 \to 6) and doubles the number shaded (2→42 \to 4), while the shaded amount does not change. That is why multiplying top and bottom by the same number is safe: it is a finer cut of the same whole, which is the same as multiplying by 22=1\tfrac22 = 1.

Example
Building equivalent fractions

Write three fractions equivalent to 23\dfrac{2}{3}.

23=2×23×2=46,23=2×33×3=69,23=2×53×5=1015.\begin{aligned} \frac{2}{3} = \frac{2\times 2}{3\times 2} = \frac{4}{6}, \qquad \frac{2}{3} = \frac{2\times 3}{3\times 3} = \frac{6}{9}, \qquad \frac{2}{3} = \frac{2\times 5}{3\times 5} = \frac{10}{15}. \end{aligned}

Each time we multiplied top and bottom by the same number, so the value did not change.

Side note
Where it goes wrong

Making an "equivalent" fraction by adding the same number to top and bottom: 23→2+13+1=34\tfrac23 \to \tfrac{2 + 1}{3 + 1} = \tfrac34. Adding changes the amount (23<34\tfrac23 < \tfrac34); only multiplying or dividing both by the same number keeps it.

Tip

Tip: Multiplying top and bottom by the same number is really multiplying by 11 in disguise (for example 22=1\tfrac{2}{2}=1). That is why the value stays the same.

Try it
Try it: missing numbers

Fill in the blanks: 38=□40\dfrac{3}{8} = \dfrac{\square}{40} and 56=20□\dfrac{5}{6} = \dfrac{20}{\square}.

Show the answer

From 88 to 4040 the denominator was multiplied by 55, so the numerator must be too: 3×5=153 \times 5 = 15, giving 1540\tfrac{15}{40}. From 55 to 2020 the numerator was multiplied by 44, so the denominator becomes 6×4=246 \times 4 = 24, giving 2024\tfrac{20}{24}.

Simplifying Fractions to Lowest Terms

Concept
Lowest Terms

A fraction is in lowest terms (fully simplified) when the numerator and denominator share no common factor except 11. To simplify, divide the top and bottom by their greatest common factor (GCF).

Simplifying regroups the pieces into bigger ones. Eighteen sixths-of-a-third can be gathered into groups of 66, making 33 large pieces, and the 1212 shaded small pieces become 22 large ones. Dividing by the GCF makes the biggest possible groups, which is why one division by the GCF lands directly in lowest terms.

Example
Simplifying 1218\dfrac{12}{18}
12 of 18 small parts = 2 of 3 large parts
Group the 1818 parts into sixes: 1218=23\tfrac{12}{18} = \tfrac23.

List factors to find the GCF of 1212 and 1818: it is 66.

1218=12÷618÷6=23.\begin{aligned} \frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}. \end{aligned}

Check: 22 and 33 share no common factor but 11, so 23\dfrac{2}{3} is fully simplified.

Side note
Where it goes wrong

Simplifying by subtracting: 1218→12−1018−10=28\tfrac{12}{18} \to \tfrac{12 - 10}{18 - 10} = \tfrac28. Only dividing both parts by a common factor keeps the value: 12÷618÷6=23\tfrac{12 \div 6}{18 \div 6} = \tfrac23.

Example
Simplifying in steps

If you do not spot the GCF right away, simplify a little at a time:

2436=24÷236÷2=1218=12÷618÷6=23.\begin{aligned} \frac{24}{36} = \frac{24 \div 2}{36 \div 2} = \frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}. \end{aligned}

You reach the same answer as long as you keep dividing until nothing but 11 divides both.

Tip

Tip: Always give final answers in lowest terms. A good habit: after every add, subtract, multiply, or divide, ask “can this simplify?”

Try it
Try it: lowest terms

Simplify 4560\dfrac{45}{60} and 2849\dfrac{28}{49}.

Show the answer

The GCF of 4545 and 6060 is 1515: 45÷1560÷15=34\tfrac{45 \div 15}{60 \div 15} = \tfrac34. The GCF of 2828 and 4949 is 77: 28÷749÷7=47\tfrac{28 \div 7}{49 \div 7} = \tfrac47.

Comparing & Ordering Fractions

Concept
Two Ways to Compare

To decide which of two fractions is larger:

  • Common denominators: rewrite both with the same denominator, then compare numerators.
  • Cross-multiplying: for ab\dfrac{a}{b} and cd\dfrac{c}{d}, compare a×da\times d with b×cb\times c. The fraction whose numerator is part of the larger product is larger.

Fractions with different denominators are cut into different-sized pieces, so their numerators cannot be compared directly. A common denominator re-cuts both wholes into the same-sized pieces, and then counting pieces is fair. Cross-multiplying is the same comparison done quickly: 35=3×840\tfrac35 = \tfrac{3 \times 8}{40} and 58=5×540\tfrac58 = \tfrac{5 \times 5}{40}, so only 2424 and 2525 need comparing.

Example
Comparing 35\dfrac{3}{5} and 58\dfrac{5}{8}
3/55/8
Same whole: 58\tfrac58 reaches a little past 35\tfrac35.

Cross-multiply: 3×8=243\times 8 = 24 and 5×5=255\times 5 = 25. Since 24<2524 < 25, the first product is smaller, so

35<58.\frac{3}{5} < \frac{5}{8}.

Check with common denominators (4040): 35=2440\dfrac{3}{5}=\dfrac{24}{40} and 58=2540\dfrac{5}{8}=\dfrac{25}{40}. Indeed 2440<2540\dfrac{24}{40}<\dfrac{25}{40}.

Side note
Where it goes wrong

Deciding by the bigger denominator: "18>16\tfrac18 > \tfrac16 because 8>68 > 6." A bigger denominator means the whole is cut into more, smaller pieces, so 18<16\tfrac18 < \tfrac16.

Example
Ordering 12, 23, 34\dfrac{1}{2},\ \dfrac{2}{3},\ \dfrac{3}{4} from least to greatest

Use the common denominator 1212:

12=612,23=812,34=912.\frac{1}{2}=\frac{6}{12},\qquad \frac{2}{3}=\frac{8}{12},\qquad \frac{3}{4}=\frac{9}{12}.

Compare numerators 6<8<96<8<9, so the order is 12<23<34\dfrac{1}{2} < \dfrac{2}{3} < \dfrac{3}{4}.

Tip

Tip: With the same numerator, the fraction with the smaller denominator is larger (bigger pieces). So 23>25\dfrac{2}{3} > \dfrac{2}{5}.

Try it
Try it: order

Order 56\dfrac{5}{6}, 79\dfrac{7}{9}, 34\dfrac{3}{4} from least to greatest.

Show the answer

The LCD of 66, 99, 44 is 3636: 56=3036\tfrac56 = \tfrac{30}{36}, 79=2836\tfrac79 = \tfrac{28}{36}, 34=2736\tfrac34 = \tfrac{27}{36}. So 34<79<56\tfrac34 < \tfrac79 < \tfrac56.

Improper Fractions & Mixed Numbers

Concept
Two Names for the Same Amount

An improper fraction has a numerator greater than or equal to its denominator (like 114\tfrac{11}{4}). A mixed number is a whole number next to a proper fraction (like 2342\tfrac{3}{4}). They can name the same amount.

  • Improper →\to mixed: divide numerator by denominator. The quotient is the whole number; the remainder over the denominator is the fraction.
  • Mixed →\to improper: multiply the whole number by the denominator, add the numerator, and keep the denominator.

Converting is regrouping by the denominator. Every 44 quarters make one whole, so 1111 quarters are 22 groups of 44 (two wholes) with 33 quarters left over, which is exactly 11÷4=211 \div 4 = 2 R 33. Going back, 3253\tfrac25 is 33 wholes of 55 fifths each plus 22 more fifths: 3×5+2=173 \times 5 + 2 = 17 fifths.

Example
Converting both ways
11 quarters = 2 wholes and 3 quarters
114=234\tfrac{11}{4} = 2\tfrac34: two full bars and three quarters of another.

Improper to mixed: 114\dfrac{11}{4}. Since 11÷4=211 \div 4 = 2 remainder 33,

114=234.\frac{11}{4} = 2\frac{3}{4}.

Mixed to improper: 3253\dfrac{2}{5}. Compute 3×5+2=173\times 5 + 2 = 17,

325=175.3\frac{2}{5} = \frac{17}{5}.
Side note
Where it goes wrong

Converting 3253\tfrac25 as 3×2+55=115\tfrac{3 \times 2 + 5}{5} = \tfrac{11}{5}. The whole number multiplies the denominator (how many fifths are in 33 wholes): 3×5+2=173 \times 5 + 2 = 17, so 175\tfrac{17}{5}.

Tip

Tip: For most multiplying and dividing, first turn every mixed number into an improper fraction. It keeps the arithmetic simple.

Try it
Try it: convert

Write 296\dfrac{29}{6} as a mixed number and 4374\dfrac{3}{7} as an improper fraction.

Show the answer

29÷6=429 \div 6 = 4 R 55, so 296=456\tfrac{29}{6} = 4\tfrac56. For 4374\tfrac37: 44 wholes are 4×7=284 \times 7 = 28 sevenths, plus 33 more is 3131, so 317\tfrac{31}{7}.

Adding & Subtracting: LIKE Denominators

Concept
Same Bottom, Add the Tops

When two fractions have the same denominator, add or subtract the numerators and keep the denominator. Then simplify.

ac+bc=a+bc,ac−bc=a−bc.\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}, \qquad \frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}.

The denominator names the size of the pieces and the numerator counts them. Three eighths plus one eighth is four of the same-sized pieces, four eighths, just as 33 apples plus 11 apple is 44 apples. The denominator stays because the pieces did not change size; adding the denominators would claim they had.

Example
Like denominators
3 eighths + 1 eighth = 4 eighths = 1/2
Same-size pieces: count them.
38+18=48=12,56−16=46=23.\frac{3}{8} + \frac{1}{8} = \frac{4}{8} = \frac{1}{2}, \qquad \frac{5}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}.

Keep the denominator the same; only the numerators change. Always simplify the result.

Side note
Where it goes wrong

Adding the denominators: 38+18=416\tfrac38 + \tfrac18 = \tfrac{4}{16}. That answer is 14\tfrac14, smaller than the 38\tfrac38 you started with, which cannot be right after adding more.

Tip

Common mistake: Do not add the denominators. 38+18\dfrac{3}{8}+\dfrac{1}{8} is 48\dfrac{4}{8}, not 416\dfrac{4}{16}.

Try it
Try it: like denominators

Compute 712+1112\dfrac{7}{12} + \dfrac{11}{12} and 910−310\dfrac{9}{10} - \dfrac{3}{10}, simplifying.

Show the answer

712+1112=1812=32=112\tfrac{7}{12} + \tfrac{11}{12} = \tfrac{18}{12} = \tfrac32 = 1\tfrac12. 910−310=610=35\tfrac{9}{10} - \tfrac{3}{10} = \tfrac{6}{10} = \tfrac35.

Adding & Subtracting: UNLIKE Denominators (LCD)

Concept
First Make the Denominators Match

To add or subtract fractions with different denominators, rewrite them over a common denominator. The easiest one to use is the least common denominator (LCD), which is the least common multiple of the denominators. Then add or subtract the numerators and simplify.

Thirds and quarters are different-sized pieces, so they cannot be counted together until both are re-cut into a common size. Twelfths work because 1212 is a multiple of both 33 and 44: each third is exactly 44 twelfths and each quarter is exactly 33 twelfths. The least common denominator is the coarsest cut that fits both, which keeps the numbers small.

Example
Adding 23+14\dfrac{2}{3}+\dfrac{1}{4}
2/31/411/12
Re-cut both into twelfths: 8+3=118 + 3 = 11 twelfths.

The LCD of 33 and 44 is 1212.

23+14=2×412+1×312=812+312=1112.\begin{aligned} \frac{2}{3} + \frac{1}{4} = \frac{2\times 4}{12} + \frac{1\times 3}{12} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}. \end{aligned}

Since 1111 and 1212 share no common factor, 1112\dfrac{11}{12} is already in lowest terms.

Side note
Where it goes wrong

Adding tops and bottoms: 23+14=37\tfrac23 + \tfrac14 = \tfrac37. Since 37\tfrac37 is less than 23\tfrac23, the "sum" is smaller than one of the parts; the pieces must be made the same size first: 812+312=1112\tfrac{8}{12} + \tfrac{3}{12} = \tfrac{11}{12}.

Example
Subtracting 56−38\dfrac{5}{6}-\dfrac{3}{8}

The LCD of 66 and 88 is 2424.

56−38=2024−924=1124.\begin{aligned} \frac{5}{6} - \frac{3}{8} = \frac{20}{24} - \frac{9}{24} = \frac{11}{24}. \end{aligned}
Tip

Tip: You need common denominators to add or subtract, but not to multiply. Multiplying fractions does not care about denominators matching.

Try it
Try it: unlike denominators

Compute 310+415\dfrac{3}{10} + \dfrac{4}{15} and 78−56\dfrac{7}{8} - \dfrac{5}{6}.

Show the answer

LCD of 1010 and 1515 is 3030: 930+830=1730\tfrac{9}{30} + \tfrac{8}{30} = \tfrac{17}{30}. LCD of 88 and 66 is 2424: 2124−2024=124\tfrac{21}{24} - \tfrac{20}{24} = \tfrac{1}{24}.

Adding & Subtracting Mixed Numbers

Concept
Wholes with Wholes, Fractions with Fractions

Add (or subtract) the whole-number parts and the fraction parts separately. Give the fractions a common denominator first. Sometimes subtracting forces you to borrow (regroup) one whole into fraction form.

A mixed number is a sum, 234=2+342\tfrac34 = 2 + \tfrac34, so the wholes and the fractions can be added separately and combined (the commutative and associative properties allow the regrouping). Borrowing trades one whole for dd\tfrac{d}{d} of the same-sized pieces, exactly like borrowing a ten in whole-number subtraction.

Example
Adding with regrouping

Compute 234+1232\dfrac{3}{4} + 1\dfrac{2}{3}. The LCD of 44 and 33 is 1212.

234+123=2912+1812=31712.\begin{aligned} 2\frac{3}{4} + 1\frac{2}{3} &= 2\frac{9}{12} + 1\frac{8}{12} = 3\frac{17}{12}. \end{aligned}

Since 1712=1512\dfrac{17}{12}=1\dfrac{5}{12}, regroup: 31712=45123\dfrac{17}{12} = 4\dfrac{5}{12}.

Side note
Where it goes wrong

Subtracting the smaller fraction from the larger: 514−2345\tfrac14 - 2\tfrac34 done as 33 and 34−14=24\tfrac34 - \tfrac14 = \tfrac24, giving 3123\tfrac12. The fraction being subtracted is the 34\tfrac34; borrowing gives 454−234=224=2124\tfrac54 - 2\tfrac34 = 2\tfrac24 = 2\tfrac12. Check: 212+234=5142\tfrac12 + 2\tfrac34 = 5\tfrac14.

Example
Subtracting with borrowing

Compute 514−2345\dfrac{1}{4} - 2\dfrac{3}{4}. You cannot take 34\dfrac{3}{4} from 14\dfrac{1}{4}, so borrow 11 whole from the 55:

514=4+1+14=454.5\frac{1}{4} = 4 + 1 + \frac{1}{4} = 4\frac{5}{4}.

Now subtract: 454−234=224=212.4\dfrac{5}{4} - 2\dfrac{3}{4} = 2\dfrac{2}{4} = 2\dfrac{1}{2}.

Tip

Tip: When you borrow one whole, it becomes a fraction equal to dd\tfrac{d}{d} (the denominator over itself). Add that to the fraction you already have.

Try it
Try it: borrow

Compute 613−2566\dfrac{1}{3} - 2\dfrac{5}{6}.

Show the answer

Common denominator 66: 626−2566\tfrac26 - 2\tfrac56. Since 26<56\tfrac26 < \tfrac56, borrow one whole: 626=5866\tfrac26 = 5\tfrac86. Then 586−256=336=3125\tfrac86 - 2\tfrac56 = 3\tfrac36 = 3\tfrac12.

Multiplying Fractions & Mixed Numbers

Concept
Straight Across

To multiply fractions, multiply the numerators and multiply the denominators. Simplify at the end (or cancel common factors first).

ab×cd=a×cb×d.\frac{a}{b}\times\frac{c}{d} = \frac{a\times c}{b\times d}.

For mixed numbers, first change each to an improper fraction, then multiply.

"23\tfrac23 of 34\tfrac34" is an area. Shade 34\tfrac34 of a square with vertical cuts, then take 23\tfrac23 of that with horizontal cuts. The cuts make 3×4=123 \times 4 = 12 small rectangles (the denominators multiplied), and the doubly shaded part is 2×3=62 \times 3 = 6 of them (the numerators multiplied). That is why no common denominator is needed: the product builds its own.

Example
Multiplying and canceling
3/42/36 of 12 squares are in both: 6/12 = 1/2
23\tfrac23 of 34\tfrac34: the overlap is 612\tfrac{6}{12}.
23×34=2×33×4=612=12.\begin{aligned} \frac{2}{3}\times\frac{3}{4} = \frac{2\times 3}{3\times 4} = \frac{6}{12} = \frac{1}{2}. \end{aligned}

You could also cancel first: the 33 on top and bottom cancel, leaving 24=12\dfrac{2}{4}=\dfrac{1}{2}.

Side note
Where it goes wrong

Multiplying mixed numbers part by part: 112×213=2161\tfrac12 \times 2\tfrac13 = 2\tfrac16 (wholes times wholes, fractions times fractions). That skips the cross terms 1×131 \times \tfrac13 and 12×2\tfrac12 \times 2. Converting first gives 32×73=312\tfrac32 \times \tfrac73 = 3\tfrac12.

Example
Multiplying mixed numbers

Compute 112×2131\dfrac{1}{2}\times 2\dfrac{1}{3}. Convert first: 112=321\dfrac{1}{2}=\dfrac{3}{2} and 213=732\dfrac{1}{3}=\dfrac{7}{3}.

32×73=216=72=312.\frac{3}{2}\times\frac{7}{3} = \frac{21}{6} = \frac{7}{2} = 3\frac{1}{2}.
Tip

Tip: “Of” usually means multiply. “12\tfrac{1}{2} of 34\tfrac{3}{4}” means 12×34=38\tfrac{1}{2}\times\tfrac{3}{4}=\tfrac{3}{8}.

Try it
Try it: cancel first

Compute 58×1225\dfrac{5}{8} \times \dfrac{12}{25} and 223×1182\dfrac{2}{3} \times 1\dfrac{1}{8}.

Show the answer

Cancel before multiplying: 55 and 2525 share 55, 88 and 1212 share 44, leaving 12×35=310\tfrac{1}{2} \times \tfrac{3}{5} = \tfrac{3}{10}. For the second, convert: 83×98\tfrac83 \times \tfrac98; the 88s cancel and 9÷3=39 \div 3 = 3, leaving 33.

Dividing Fractions & Mixed Numbers

Concept
Multiply by the Reciprocal

To divide by a fraction, multiply by its reciprocal (flip the second fraction). The reciprocal of cd\dfrac{c}{d} is dc\dfrac{d}{c}.

ab÷cd=ab×dc.\frac{a}{b}\div\frac{c}{d} = \frac{a}{b}\times\frac{d}{c}.

Change any mixed numbers to improper fractions before flipping.

Division asks "how many of these fit into that?" How many 1141\tfrac14s fit into 2122\tfrac12? Exactly 22, as the jumps show. The reciprocal rule gets there in general: dividing by cd\tfrac{c}{d} must undo multiplying by cd\tfrac{c}{d}, and multiplying by dc\tfrac{d}{c} does exactly that, since cd×dc=1\tfrac{c}{d} \times \tfrac{d}{c} = 1.

Example
Dividing fractions
34÷25=34×52=158=178.\begin{aligned} \frac{3}{4}\div\frac{2}{5} = \frac{3}{4}\times\frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}. \end{aligned}
Side note
Where it goes wrong

Flipping the first fraction: 34÷25\tfrac34 \div \tfrac25 done as 43×25=815\tfrac43 \times \tfrac25 = \tfrac{8}{15}. Only the divisor is flipped: 34×52=158\tfrac34 \times \tfrac52 = \tfrac{15}{8}. Sense check: 25\tfrac25 fits into 34\tfrac34 almost twice, and 158\tfrac{15}{8} is almost 22.

Example
Dividing mixed numbers
01231 1/41 1/42 1/2
Two jumps of 1141\tfrac14 reach 2122\tfrac12: the quotient is 22.

Compute 212÷1142\dfrac{1}{2}\div 1\dfrac{1}{4}. Convert: 212=522\dfrac{1}{2}=\dfrac{5}{2} and 114=541\dfrac{1}{4}=\dfrac{5}{4}.

52÷54=52×45=2010=2.\frac{5}{2}\div\frac{5}{4} = \frac{5}{2}\times\frac{4}{5} = \frac{20}{10} = 2.
Tip

Tip: Only flip the fraction after the division sign (the divisor). Flipping the wrong one is the most common division mistake.

Try it
Try it: how many fit?

How many 38\tfrac{3}{8}-cup scoops are in 4124\tfrac{1}{2} cups? Then compute 23÷4\dfrac{2}{3} \div 4.

Show the answer

412÷38=92×83=726=124\tfrac12 \div \tfrac38 = \tfrac92 \times \tfrac83 = \tfrac{72}{6} = 12 scoops. For 23÷4\tfrac23 \div 4, the reciprocal of 44 is 14\tfrac14: 23×14=212=16\tfrac23 \times \tfrac14 = \tfrac{2}{12} = \tfrac16.

Word Problems with Fractions

Concept
A Plan for Word Problems
  • Read carefully and decide what operation the story describes.
  • “Of” a quantity means multiply; “left over” or “how much more” often means subtract; “in all” means add; sharing equally means divide.
  • Do the arithmetic, then simplify and label your answer.

The operation comes from the relationship in the story, not the numbers. "Half of" something is a scaling, so it is multiplication, even though the answer is smaller; "how many portions fit" is division, even though the answer is larger. Fractions make these surprises common, so an estimate first (should the answer be more or less than what I started with?) is the best check.

Example
A recipe problem
half of 3/4 cup = 3/8 cup
Cutting each quarter in half: 34\tfrac34 becomes 68\tfrac68, and half of that is 38\tfrac38.

A recipe needs 34\dfrac{3}{4} cup of flour. You are making half a batch. How much flour do you need?

Half of 34\dfrac{3}{4} means 12×34=38\dfrac{1}{2}\times\dfrac{3}{4} = \dfrac{3}{8} cup of flour.

Side note
Where it goes wrong

Dividing by 12\tfrac12 for "half of 34\tfrac34": 34÷12=112\tfrac34 \div \tfrac12 = 1\tfrac12 cups, twice the original. Taking half makes less, so the operation is multiplying by 12\tfrac12: 38\tfrac38 cup.

Example
A distance problem

Maya jogged 2122\dfrac{1}{2} miles on Monday and 1341\dfrac{3}{4} miles on Tuesday. How far in all?

212+134=224+134=354=414 miles.\begin{aligned} 2\frac{1}{2} + 1\frac{3}{4} = 2\frac{2}{4} + 1\frac{3}{4} = 3\frac{5}{4} = 4\frac{1}{4}\text{ miles.} \end{aligned}
Tip

Tip: Estimate first. 212+1342\tfrac12 + 1\tfrac34 should be a bit more than 44, so the answer 4144\tfrac14 is reasonable. Estimating catches big mistakes.

Try it
Try it: a ribbon

A ribbon 5145\tfrac14 m long is cut into pieces 34\tfrac34 m long. How many pieces? What length is 23\tfrac23 of the original ribbon?

Show the answer

Pieces: 514÷34=214×43=75\tfrac14 \div \tfrac34 = \tfrac{21}{4} \times \tfrac43 = 7 pieces. Two-thirds of it: 23×214=4212=312\tfrac23 \times \tfrac{21}{4} = \tfrac{42}{12} = 3\tfrac12 m.

Going Deeper: Advanced Fraction Ideas

Concept
Complex Fractions (a Fraction of Fractions)

A complex fraction is a fraction whose numerator, denominator, or both are themselves fractions, like  23  45 \dfrac{\ \frac{2}{3}\ }{\ \frac{4}{5}\ }. A big fraction bar means “divide,” so a complex fraction is just a division problem in disguise:

 ab  cd =ab÷cd=ab×dc.\frac{\ \frac{a}{b}\ }{\ \frac{c}{d}\ } = \frac{a}{b}\div\frac{c}{d} = \frac{a}{b}\times\frac{d}{c}.

Rewrite it as a division, multiply by the reciprocal, and simplify as usual.

Example
Simplifying a complex fraction

Simplify  23  45 \dfrac{\ \frac{2}{3}\ }{\ \frac{4}{5}\ }. Read the main bar as “divide”:

 23  45 =23÷45=23×54=1012=56.\begin{aligned} \frac{\ \frac{2}{3}\ }{\ \frac{4}{5}\ } = \frac{2}{3}\div\frac{4}{5} = \frac{2}{3}\times\frac{5}{4} = \frac{10}{12} = \frac{5}{6}. \end{aligned}

So a stack of two fractions collapses to the single fraction 56\dfrac{5}{6}.

Concept
Why Cross-Multiplication Works — and the Mediant

When you compare ab\dfrac{a}{b} and cd\dfrac{c}{d} (with positive bb and dd), multiplying both by bdbd does not change which is larger, since bd>0bd>0. This turns ab\dfrac{a}{b} into adad and cd\dfrac{c}{d} into bcbc — exactly the cross-products. That is the whole justification for the cross-multiplying rule.

A surprising cousin is the mediant of ab\dfrac{a}{b} and cd\dfrac{c}{d}, formed by adding tops and bottoms:

mediant=a+cb+d.\text{mediant} = \frac{a+c}{b+d}.

For positive fractions, the mediant always lands strictly between the two original fractions. (Warning: this is not how you add fractions — it is a different, special construction.)

Concept
Unit Fractions and Egyptian Fractions

A unit fraction has a numerator of 11, like 12\dfrac{1}{2} or 17\dfrac{1}{7}. The ancient Egyptians wrote every other fraction as a sum of distinct unit fractions, for example 23=12+16\dfrac{2}{3}=\dfrac{1}{2}+\dfrac{1}{6}.

One reliable recipe is the greedy method: repeatedly subtract the largest unit fraction that is not bigger than what remains, until nothing is left. It always terminates, and the denominators grow quickly.

Example
An Egyptian fraction for 37\dfrac{3}{7}

Write 37\dfrac{3}{7} as a sum of distinct unit fractions using the greedy method.

Step 1. The largest unit fraction ≤37\le \dfrac{3}{7} is 13\dfrac{1}{3} (since 12>37\dfrac{1}{2}>\dfrac{3}{7}). Subtract it:

37−13=921−721=221.\frac{3}{7}-\frac{1}{3} = \frac{9}{21}-\frac{7}{21} = \frac{2}{21}.

Step 2. The largest unit fraction ≤221\le \dfrac{2}{21} is 111\dfrac{1}{11} (since 110>221\dfrac{1}{10}>\dfrac{2}{21}). Subtract it:

221−111=22231−21231=1231.\frac{2}{21}-\frac{1}{11} = \frac{22}{231}-\frac{21}{231} = \frac{1}{231}.

The remainder is already a unit fraction, so we stop:

37=13+111+1231.\frac{3}{7} = \frac{1}{3}+\frac{1}{11}+\frac{1}{231}.

Check: 13+111=1433=98231\dfrac{1}{3}+\dfrac{1}{11}=\dfrac{14}{33}=\dfrac{98}{231}, and 98231+1231=99231=37\dfrac{98}{231}+\dfrac{1}{231}=\dfrac{99}{231}=\dfrac{3}{7}. ✓

Concept
Continued Fractions

A continued fraction builds a number out of a whole part plus 11 over another whole part plus 11 over …\ldots, stacking reciprocals downward. Every ordinary fraction can be rewritten this way by repeatedly separating the whole-number part and flipping the leftover. For example,

75=1+25=1+1 52 =1+12+12.\frac{7}{5} = 1 + \frac{2}{5} = 1 + \cfrac{1}{\ \frac{5}{2}\ } = 1 + \cfrac{1}{2 + \cfrac{1}{2}}.

Reading it back from the bottom up: 2+12=522+\tfrac12=\tfrac52, then 1÷52=251\div\tfrac52=\tfrac25, then 1+25=751+\tfrac25=\tfrac75. Continued fractions reveal the “best” simple approximations to a number.

Concept
The Harmonic Mean

The harmonic mean of two positive numbers aa and bb averages their reciprocals instead of the numbers themselves:

H=2 1a+1b =2aba+b.H = \frac{2}{\ \frac{1}{a}+\frac{1}{b}\ } = \frac{2ab}{a+b}.

It is the right average when the quantities are rates. If you drive one mile at 3030 mph and one mile at 6060 mph, your average speed is the harmonic mean

2×30×6030+60=360090=40 mph,\frac{2\times 30\times 60}{30+60} = \frac{3600}{90} = 40 \text{ mph,}

which is less than the plain average of 4545, because you spend more time at the slower speed.

Example
A telescoping sum

Add 11⋅2+12⋅3+13⋅4+14⋅5\dfrac{1}{1\cdot 2}+\dfrac{1}{2\cdot 3}+\dfrac{1}{3\cdot 4}+\dfrac{1}{4\cdot 5}.

The key trick is that each term splits into a difference of unit fractions:

1n(n+1)=1n−1n+1.\frac{1}{n(n+1)} = \frac{1}{n}-\frac{1}{n+1}.

For instance 12⋅3=12−13\dfrac{1}{2\cdot 3}=\dfrac{1}{2}-\dfrac{1}{3}. Writing the whole sum this way, the inside terms cancel in pairs (this is called telescoping):

(11−12)+(12−13)+(13−14)+(14−15)=1−15=45.\begin{aligned} \left(\frac{1}{1}-\frac{1}{2}\right) +\left(\frac{1}{2}-\frac{1}{3}\right) +\left(\frac{1}{3}-\frac{1}{4}\right) +\left(\frac{1}{4}-\frac{1}{5}\right) = 1 - \frac{1}{5} = \frac{4}{5}. \end{aligned}

Only the very first and very last pieces survive. In general such a sum up to 1N(N+1)\dfrac{1}{N(N+1)} equals NN+1\dfrac{N}{N+1}.

Why It Works

Cross-multiplication compares ab\tfrac ab and cd\tfrac cd correctly because multiplying both by the positive number bdbd preserves order and turns the fractions into the whole numbers adad and bcbc. The reciprocal rule for division works because ab⋅ba=1\tfrac ab\cdot\tfrac ba = 1, so multiplying by ba\tfrac ba undoes multiplying by ab\tfrac ab. Every fraction rule is a consequence of "a fraction is a division", and can be re-derived from that in a line.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.

Formulas, Proofs & Tips

Tip
Fraction arithmetic
ab±cd=ad±bcbd,ab⋅cd=acbd,ab÷cd=ab⋅dc\frac{a}{b}\pm\frac{c}{d}=\frac{ad\pm bc}{bd},\qquad \frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd},\qquad \frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}

What it means. Add with a common denominator, multiply straight across, divide by flipping the second fraction.

Example. 23+16=46+16=56\tfrac23+\tfrac16=\tfrac46+\tfrac16=\tfrac56.

Why it works. Rewriting over the common denominator bdbd makes the pieces the same size so they can be counted together. Division asks "how many cd\tfrac{c}{d} fit?", and multiplying by the reciprocal answers it because cd⋅dc=1\tfrac{c}{d}\cdot\tfrac{d}{c}=1.

Tip. Simplify before multiplying — cancelling early keeps the numbers small.