Intuition The Big PictureA fraction, a decimal, and a percentage are three different costumes worn by the same number . 1 2 \frac{1}{2} 2 1 0.5 0.5 0.5 50 % 50\% 50% same point on the number line. Converting between them is just changing the costume, not the person .
WHY do we have three forms? Because each is convenient in a different situation. Fractions for exact ratios (1 3 \frac{1}{3} 3 1 0.333 … 0.333\ldots 0.333 … 33 % 33\% 33% WHAT unifies them? They all answer "how much of one whole?" A percent literally means "per hundred" — so the whole is chopped into 100 pieces.
A fraction a b \frac{a}{b} b a a a a b b b b ≠ 0 b \neq 0 b = 0
A decimal writes a number using place values: tenths, hundredths, thousandths (0.1 = 1 10 0.1 = \frac{1}{10} 0.1 = 10 1 0.01 = 1 100 0.01 = \frac{1}{100} 0.01 = 100 1
A percentage p % p\% p % p 100 \frac{p}{100} 100 p p p p
The single fact that makes everything click:
p % = p 100 p\% = \frac{p}{100} p % = 100 p
WHY this is the master key: The symbol "% \% % ÷ 100 \div 100 ÷ 100 % \% % ÷ 100 \div 100 ÷ 100 becomes a fraction, which you can then turn into a decimal. Every conversion below is built from this one idea.
A fraction a b \frac{a}{b} b a is the division a ÷ b a \div b a ÷ b do the long division .
WHY: 3 4 \frac{3}{4} 4 3 means "3 divided into 4 equal shares". Division is the definition, not a trick.
3 4 = 3 ÷ 4 = 0.75 \frac{3}{4} = 3 \div 4 = 0.75 4 3 = 3 ÷ 4 = 0.75
Read the place value of the last digit and put it over the matching power of 10, then simplify.
WHY: 0.75 0.75 0.75 = 75 100 = \frac{75}{100} = 100 75 ⇒ \Rightarrow ⇒ 100 100 100 0.75 = 75 100 = 75 ÷ 25 100 ÷ 25 = 3 4 0.75 = \frac{75}{100} = \frac{75 \div 25}{100 \div 25} = \frac{3}{4} 0.75 = 100 75 = 100 ÷ 25 75 ÷ 25 = 4 3
Since p % = p 100 p\% = \frac{p}{100} p % = 100 p p p p "how many hundredths is this decimal?" — multiply by 100.
decimal × 100 = percent \text{decimal} \times 100 = \text{percent} decimal × 100 = percent WHY: 0.75 = 75 100 = 75 % 0.75 = \frac{75}{100} = 75\% 0.75 = 100 75 = 75%
Undo the above: divide by 100 (i.e. replace % \% % ÷ 100 \div 100 ÷ 100 75 % = 75 100 = 0.75 75\% = \frac{75}{100} = 0.75 75% = 100 75 = 0.75
Combine: turn the fraction into a decimal, then × 100 \times 100 × 100 3 4 = 3 × 25 4 × 25 = 75 100 = 75 % \frac{3}{4} = \frac{3 \times 25}{4 \times 25} = \frac{75}{100} = 75\% 4 3 = 4 × 25 3 × 25 = 100 75 = 75%
p % = p 100 p\% = \frac{p}{100} p % = 100 p 40 % = 40 100 = 2 5 40\% = \frac{40}{100} = \frac{2}{5} 40% = 100 40 = 5 2
Worked example Example 1 —5 8 \frac{5}{8} 8 5 Step 1: 5 ÷ 8 = 0.625 5 \div 8 = 0.625 5 ÷ 8 = 0.625 Why this step? A fraction is a division; long division gives the decimal.
Step 2: 0.625 × 100 = 62.5 % 0.625 \times 100 = 62.5\% 0.625 × 100 = 62.5% Why this step? Percent = number of hundredths, so multiply by 100.
Answer: 5 8 = 0.625 = 62.5 % \frac{5}{8} = 0.625 = 62.5\% 8 5 = 0.625 = 62.5%
Worked example Example 2 —0.35 0.35 0.35 Step 1: Last digit (5 5 5 hundredths place ⇒ 35 100 \Rightarrow \frac{35}{100} ⇒ 100 35 Why this step? Place value tells you the denominator.
Step 2: Simplify: gcd ( 35 , 100 ) = 5 ⇒ 7 20 \gcd(35,100)=5 \Rightarrow \frac{7}{20} g cd( 35 , 100 ) = 5 ⇒ 20 7 Why this step? Fractions should be in lowest terms.
Step 3: 0.35 × 100 = 35 % 0.35 \times 100 = 35\% 0.35 × 100 = 35% Answer: 0.35 = 7 20 = 35 % 0.35 = \frac{7}{20} = 35\% 0.35 = 20 7 = 35%
Worked example Example 3 — Recurring decimal1 3 \frac{1}{3} 3 1 Step 1: 1 ÷ 3 = 0.3333 … = 0. 3 ‾ 1 \div 3 = 0.3333\ldots = 0.\overline{3} 1 ÷ 3 = 0.3333 … = 0. 3 Why this step? The long division never reaches remainder 0.
Step 2: 0. 3 ‾ × 100 = 33. 3 ‾ % = 33 1 3 % 0.\overline{3} \times 100 = 33.\overline{3}\% = 33\tfrac13\% 0. 3 × 100 = 33. 3 % = 33 3 1 % Why this step? Same × 100 \times100 × 100 Answer: 1 3 = 0. 3 ‾ = 33 1 3 % \frac{1}{3} = 0.\overline{3} = 33\tfrac13\% 3 1 = 0. 3 = 33 3 1 %
Worked example Example 4 — A percentageover 100: 150 % 150\% 150% Step 1: 150 % = 150 100 = 1.5 150\% = \frac{150}{100} = 1.5 150% = 100 150 = 1.5 Why this step? Replace % \% % ÷ 100 \div100 ÷ 100 Step 2: 1.5 = 3 2 1.5 = \frac{3}{2} 1.5 = 2 3 Answer: 150 % = 1.5 = 3 2 150\% = 1.5 = \frac{3}{2} 150% = 1.5 = 2 3
Common mistake "To convert a decimal to a percent, just move the point — but which way?"Why the wrong way feels right: Students remember "move the decimal point 2 places" but forget the direction, so they turn 0.5 0.5 0.5 0.5 % 0.5\% 0.5% 5 % 5\% 5% The fix: Anchor to meaning, not motion. 0.5 = 50 100 = 50 % 0.5 = \frac{50}{100} = 50\% 0.5 = 100 50 = 50% always multiply by 100 going to percent, divide by 100 going from percent. 0.5 0.5 0.5 half — half is obviously 50 % 50\% 50% 5 % 5\% 5%
1 4 = 1.4 % \frac{1}{4} = 1.4\% 4 1 = 1.4% Why it feels right: The digits 1 and 4 are just... reused.
The fix: A fraction is a division first: 1 ÷ 4 = 0.25 1\div 4 = 0.25 1 ÷ 4 = 0.25 × 100 = 25 % \times 100 = 25\% × 100 = 25%
1 3 \frac{1}{3} 3 1 Why it feels right: 0.33 0.33 0.33 The fix: 0.33 = 33 % 0.33 = 33\% 0.33 = 33% approximate . Exact is 33 1 3 % 33\tfrac13\% 33 3 1 %
Common mistake Wrong denominator from place valueWhy it feels right: 0.7 0.7 0.7 looks like "7 out of 100".
The fix: Count places. One decimal place ⇒ \Rightarrow ⇒ ⇒ 7 10 \Rightarrow \frac{7}{10} ⇒ 10 7 not 7 100 \frac{7}{100} 100 7
Recall Feynman: explain it to a 12-year-oldImagine a chocolate bar. If you break it into 4 equal chunks and eat 3 , that's the fraction 3 4 \frac34 4 3 100 tiny squares, you'd have eaten 75 of them — that's the percentage , 75 % 75\% 75% 3 ÷ 4 = 0.75 3 \div 4 = 0.75 3 ÷ 4 = 0.75 decimal . Same amount of chocolate, three ways of saying it! To swap between them: dividing gives you the decimal, and "× 100 \times 100 × 100 ÷ 100 \div 100 ÷ 100
Mnemonic Remember the direction"To Percent, Push it UP (×100); From Percent, Fall Down (÷100)."
And for the ring: "Fractions Divide into Decimals, Decimals ×100 into Percent." (F→D: D ivide; D→P: P lus two zeros' worth = ×100.)
Fraction
Decimal
Percent
1 2 \frac12 2 1 0.5 0.5 0.5 50 % 50\% 50%
1 4 \frac14 4 1 0.25 0.25 0.25 25 % 25\% 25%
3 4 \frac34 4 3 0.75 0.75 0.75 75 % 75\% 75%
1 5 \frac15 5 1 0.2 0.2 0.2 20 % 20\% 20%
1 8 \frac18 8 1 0.125 0.125 0.125 12.5 % 12.5\% 12.5%
1 3 \frac13 3 1 0. 3 ‾ 0.\overline3 0. 3 33 1 3 % 33\tfrac13\% 33 3 1 %
1 100 \frac{1}{100} 100 1 0.01 0.01 0.01 1 % 1\% 1%
80/20: Memorise this table's first six rows. They cover the vast majority of exam conversions instantly.
What does the symbol % \% % "per hundred", i.e.
÷ 100 \div 100 ÷ 100 .
Convert 3 4 \frac{3}{4} 4 3 3 ÷ 4 = 0.75 3 \div 4 = 0.75 3 ÷ 4 = 0.75 .
Convert 0.75 0.75 0.75 0.75 × 100 = 75 % 0.75 \times 100 = 75\% 0.75 × 100 = 75% .
How do you turn a decimal into a percentage? Multiply by 100.
How do you turn a percentage into a decimal? Divide by 100.
Convert 40 % 40\% 40% 40 100 = 2 5 \frac{40}{100} = \frac{2}{5} 100 40 = 5 2 .
Why does 0.7 = 7 10 0.7 = \frac{7}{10} 0.7 = 10 7 7 100 \frac{7}{100} 100 7 One decimal place = tenths, so denominator is 10.
Express 1 3 \frac{1}{3} 3 1 33 1 3 % 33\tfrac13\% 33 3 1 % (since
0. 3 ‾ × 100 0.\overline3 \times 100 0. 3 × 100 ).
Convert 150 % 150\% 150% 1.5 = 3 2 1.5 = \frac{3}{2} 1.5 = 2 3 (percentages can exceed 100%).
Convert 5 8 \frac{5}{8} 8 5 5 ÷ 8 = 0.625 ⇒ 62.5 % 5\div8 = 0.625 \Rightarrow 62.5\% 5 ÷ 8 = 0.625 ⇒ 62.5% .
place value over power of 10
write over 100 then simplify
Same point on number line
Intuition Hinglish mein samjhoDekho, fraction, decimal aur percentage — teeno ek hi number ke alag-alag "kapde" hain. Jaise 3 4 \frac{3}{4} 4 3 0.75 0.75 0.75 75 % 75\% 75%
Ab magic key yaad rakho: "percent" ka matlab hi hota hai "out of 100", yaani ÷ 100 \div 100 ÷ 100 % \% % ÷ 100 \div 100 ÷ 100 divide kar do (3 ÷ 4 = 0.75 3 \div 4 = 0.75 3 ÷ 4 = 0.75 × 100 \times 100 × 100 0.75 × 100 = 75 % 0.75 \times 100 = 75\% 0.75 × 100 = 75% ÷ 100 \div 100 ÷ 100
Common galti: students point ko idhar-udhar move kar dete hain bina direction sochte. Trick ye hai — meaning se socho. 0.5 0.5 0.5 aadha hai, aur aadha obviously 50 % 50\% 50% 5 % 5\% 5% 0.7 0.7 0.7 7 10 \frac{7}{10} 10 7 7 100 \frac{7}{100} 100 7
Ye topic exams me, shopping discounts me (40 % 40\% 40% 1 2 , 1 4 , 3 4 , 1 5 , 1 8 , 1 3 \frac12, \frac14, \frac34, \frac15, \frac18, \frac13 2 1 , 4 1 , 4 3 , 5 1 , 8 1 , 3 1
