Converting - fractions ↔ decimals ↔ percentages: Exercises

Figure — Converting - fractions ↔ decimals ↔ percentages

Intuition Reading the map above (accessible description)

This figure is the Conversion Ring, and every solution below walks its arrows. If you cannot see the image, here is the whole picture in words:

Three labelled circles are the three costumes of one number: the orange Fraction $\frac{a}{b}$ on the left, the teal Decimal at the top, the plum Percent $p %$ on the right. They form a shallow triangle.
Orange arrow, left circle → top circle (Fraction → Decimal): labelled "divide $a \div b$ " — turn a fraction into a decimal by dividing top by bottom.
Teal arrow, top circle → left circle (Decimal → Fraction): labelled "place value over a power of 10" — read the last digit's place, write it over $10$ , $100$ , $1000$ …, then simplify.
Plum arrow, top circle → right circle (Decimal → Percent): labelled " $\times 100$ " — count how many hundredths.
Teal arrow, right circle → top circle (Percent → Decimal): labelled " $\div 100$ " — undo the percent.
A box across the bottom states the master key $p % = \frac{p}{100}$ , the single fact that powers all four arrows.

When a solution says "Arrow used", find that arrow on this map (by colour and direction) and ask "which costume am I leaving, which am I arriving at, and why?" Every solution below names its arrow so you can trace it here.

Level 1 — Recognition

(Can you name the same number in another costume? One arrow, no chains.)

Recall Solution L1.1

Arrow used: the orange arrow (Fraction → Decimal, divide $a \div b$ ), then the plum arrow (Decimal → Percentage, $\times 100$ ). WHAT: $\frac{7}{10}$ means "7 parts out of 10". Divide: $7 \div 10 = 0.7$ . WHY: a fraction is a division — this is the definition, not a trick. Now $0.7 \times 100 = 70$ , so it is $70%$ . WHY $\times 100$ : "percent" asks how many hundredths? — and $0.7 = \frac{70}{100}$ , which is 70 hundredths. Answer: $\frac{7}{10} = 0.7 = 70%$ .

Recall Solution L1.2

Arrow used: the teal arrow (Decimal → Fraction, place value). WHAT: the last digit $4$ sits one place after the point, so it is in the tenths place $\Rightarrow \frac{4}{10}$ . WHY one zero, not two: one decimal place means denominator $10$ ; the number of digits after the point = number of zeros in the denominator (see Decimal place value). Simplify: $g cd (4, 10) = 2$ , so $\frac{4}{10} = \frac{2}{5}$ (see Fractions - simplifying and equivalent fractions). Answer: $0.4 = \frac{2}{5}$ .

Recall Solution L1.3

Arrow used: the teal arrow (Percentage → Decimal, $\div 100$ ). WHAT: replace the " $%$ " with " $\div 100$ ": $25% = \frac{25}{100} = 0.25$ . WHY: the whole point of $%$ is "out of 100", so $25%$ is literally $\frac{25}{100}$ , which is $0.25$ in place-value form (2 tenths + 5 hundredths). Answer: $25% = 0.25$ .

Level 2 — Application

(Now the numbers don't divide as cleanly. Do the actual long division / place-value work.)

Recall Solution L2.1

Arrow used: the orange arrow then the plum arrow (Fraction → Decimal → Percentage). WHAT (long division): $7 \div 8$ . Since $7 < 8$ , write $7.000$ and divide (see Long division): $70 \div 8 = 8$ remainder $6$ ; bring down: $60 \div 8 = 7$ remainder $4$ ; bring down: $40 \div 8 = 5$ remainder $0$ . Stop. So $7 \div 8 = 0.875$ . WHY it stops: the remainder hit $0$ , so this is a terminating decimal. To percent: $0.875 \times 100 = 87.5$ , i.e. $87.5%$ . WHY $\times 100$ : the plum arrow asks how many hundredths? — and $0.875 = \frac{87.5}{100}$ , i.e. $87.5$ hundredths. Answer: $\frac{7}{8} = 0.875 = 87.5%$ .

Recall Solution L2.2

Arrow used: the teal arrow (Decimal → Fraction, place value + simplify). WHAT: the last digit ( $5$ ) is three places after the point $\Rightarrow$ thousandths $\Rightarrow \frac{625}{1000}$ . Simplify: $g cd (625, 1000) = 125$ , so $\frac{625 \div 125}{1000 \div 125} = \frac{5}{8}$ . WHY 1000: three decimal places ⇒ three zeros in the denominator. Answer: $0.625 = \frac{5}{8}$ .

Recall Solution L2.3

Arrow used: Way A rescales the fraction directly to $/100$ ; Way B walks the orange then plum arrows (Fraction → Decimal → Percentage). Way A (scale the denominator to 100): $20 \times 5 = 100$ , so multiply top and bottom by 5: $\frac{9}{20} = \frac{9 \times 5}{20 \times 5} = \frac{45}{100} = 45%.$ WHY this works: a percentage is a fraction with denominator 100, so if you can rescale to $/100$ you can read the percent off directly. Way B (divide then $\times 100$ ): $9 \div 20 = 0.45$ , then $0.45 \times 100 = 45%$ . Same answer — good, they must agree because it's the same number in different costumes. Answer: $\frac{9}{20} = 45%$ .

Level 3 — Analysis

(Recurring decimals and "keep it exact" — decide what to round and what to keep as a fraction.)

Recall Solution L3.1

Arrow used: the orange arrow (Fraction → Decimal, long division), then the plum arrow (Decimal → Percentage, $\times 100$ ). WHAT (long division): $2 \div 3$ : $20 \div 3 = 6$ r $2$ ; the remainder $2$ returns forever, so digit $6$ repeats: $2 \div 3 = 0.6666 \dots = 0. \overline{6} .$ (The bar over the $6$ means " $6$ repeats forever" — see the overline definition at the top.) WHY it never stops: the remainder never reaches $0$ — it's a recurring decimal (see Recurring decimals to fractions). To percent: $0. \overline{6} \times 100 = 66. \overline{6} %$ . Keep it exact by writing the leftover third as a fraction: $66 \frac{2}{3} %$ . WHY $\times 100$ : the plum arrow counts hundredths — $0. \overline{6} = \frac{66. 6}{100}$ , i.e. $66. \overline{6}$ hundredths, and $0. \overline{6} = \frac{2}{3}$ of a hundredth-count gives the exact $66 \frac{2}{3}$ . WHY not just $67%$ : $67%$ is a rounded approximation; when a question says "exact", carry the $\frac{2}{3}$ . Answer: $\frac{2}{3} = 0. \overline{6} = 66 \frac{2}{3} %$ .

Recall Solution L3.2

Arrow used: the orange arrow (Fraction → Decimal, long division), then the plum arrow (Decimal → Percentage, $\times 100$ ). WHAT (long division): $5 \div 6$ : $50 \div 6 = 8$ r $2$ ; then $20 \div 6 = 3$ r $2$ ; the $2$ repeats, so the $3$ repeats: $5 \div 6 = 0.8333 \dots = 0.8 \overline{3} .$ (Only the $3$ carries a bar: the $8$ happens once, then $3$ loops forever.) To percent: $0.8 \overline{3} \times 100 = 83. \overline{3} % = 83 \frac{1}{3} %$ . WHY $\times 100$ : the plum arrow counts hundredths — $0.8 \overline{3} = \frac{83. 3}{100}$ , i.e. $83. \overline{3}$ hundredths, and the tail $0. \overline{3}$ is exactly $\frac{1}{3}$ . WHY the $\frac{1}{3}$ : the tail $0. \overline{3}$ is $\frac{1}{3}$ , and $80 + 3. \overline{3} = 83 \frac{1}{3}$ . Answer: $\frac{5}{6} = 0.8 \overline{3} = 83 \frac{1}{3} %$ .

Recall Solution L3.3

Arrow used: the orange then plum arrows to put $\frac{5}{8}$ into percent form, so both quantities share one costume. Strategy: put both in the same costume — percentages — so they can be compared on one number line. $\frac{5}{8} = 5 \div 8 = 0.625 = 62.5%$ . Compare with $62%$ . $62.5% > 62%$ , so $\frac{5}{8}$ is larger, by $62.5% - 62% = 0.5%$ . WHY convert first: you can't safely compare a fraction against a percentage while they wear different costumes; a common form removes the guesswork. Answer: $\frac{5}{8}$ is larger by $0.5%$ .

Level 4 — Synthesis

(Chain several arrows, or convert word problems into ring-hops.)

Recall Solution L4.1

Arrow used: Amina rescales fraction → $/100$ → decimal (teal-style read-off); Beto walks the teal arrow (Percent → Decimal, $\div 100$ ); Carla is already a decimal. All three land in decimal form. Strategy: convert all three to decimals (one common costume), then order them.

Amina: $\frac{17}{25}$ . Scale to 100: $25 \times 4 = 100$ , so $\frac{17 \times 4}{25 \times 4} = \frac{68}{100}$ . WHY $\frac{68}{100}$ is exactly $0.68$ : a denominator of $100$ means "68 hundredths", and hundredths are precisely the second place after the decimal point (see Decimal place value). So $\frac{68}{100} = 0.68$ directly — no division needed once the denominator is a power of 10.
Beto: $70% = \frac{70}{100} = 0.70$ .
Carla: already $0.68$ . Compare: $0.70 > 0.68$ , and Amina $= 0.68 =$ Carla. So Beto is highest; Amina and Carla are tied. Answer: Beto ( $0.70$ ) $>$ Amina $=$ Carla ( $0.68$ each). Amina and Carla scored exactly the same — a nice reminder that $\frac{17}{25}$ and $0.68$ are the same number in two costumes.

Recall Solution L4.2

Arrow used: the teal arrow (Percentage → Decimal, $\div 100$ ) turns $35%$ into a decimal multiplier before any arithmetic. Convert first: $35% = \frac{35}{100} = 0.35$ . WHAT it means: the discount is $0.35$ of the price, so you keep $1 - 0.35 = 0.65$ of it. Compute: new price $= 0.65 \times 40 = 26$ . WHY the $0.65$ shortcut: subtracting $35%$ once is the same as multiplying by "the fraction that survives", $65% = 0.65$ . This is exactly the decimal-costume of a percentage doing work. Answer: the shirt now costs $26.

Recall Solution L4.3

Arrow used: the algebra trick plays the teal arrow (Decimal → Fraction) for a recurring decimal, then the plum arrow (Fraction/Decimal → Percentage, $\times 100$ ). Method (the algebra trick, see Recurring decimals to fractions): let $x = 0. \overline{45}$ . The block " $45$ " has 2 digits, so multiply by $1 0^{2} = 100$ : $100 x = 45. \overline{45} .$ Subtract the original $x = 0. \overline{45}$ : $100 x - x = 45. \overline{45} - 0. \overline{45} \Rightarrow 99 x = 45.$ WHY subtract: the endless tails are identical, so they cancel, leaving a clean whole number. So $x = \frac{45}{99}$ . Simplify: $g cd (45, 99) = 9 \Rightarrow \frac{5}{11}$ . To percent (using the simplified fraction): walk the plum arrow — $\frac{5}{11} \times 100 = \frac{500}{11}$ percent. Now turn that improper fraction into a mixed number: $500 \div 11 = 45$ remainder $5$ , so $\frac{500}{11} = 45 \frac{5}{11}$ . Hence $45 \frac{5}{11} %$ . WHY use $\frac{5}{11}$ , not $\frac{45}{99}$ : they are the same number, so both give the same percent; we prefer the simplified $\frac{5}{11}$ so the arithmetic ( $5 \times 100 = 500$ ) stays small and the final mixed number is already in lowest terms. Answer: $0. \overline{45} = \frac{5}{11} = 45 \frac{5}{11} %$ .

Level 5 — Mastery

(Reverse-engineering, unknowns, edge cases beyond 100%, and the full round-trip.)

Recall Solution L5.1

Arrow used: work the ring backwards — Percentage → Decimal (teal, $\div 100$ ) → Fraction (teal, place value + simplify). WHAT: $37.5% = \frac{37.5}{100}$ . This numerator still has a decimal point in it, which is not allowed in a proper fraction — we need whole numbers top and bottom. Clear the decimal by multiplying by $\frac{10}{10}$ : $\frac{37.5}{100} = \frac{37.5}{100} \times \frac{10}{10} = \frac{375}{1000} .$ WHY $\frac{10}{10}$ works: $\frac{10}{10} = 1$ , so multiplying by it does not change the value — it only changes the costume. Multiplying the top $37.5$ by $10$ slides its decimal point one place right to give the whole number $375$ ; multiplying the bottom by $10$ keeps the fraction balanced. (Choose $10$ because there is exactly one digit after the point; two digits would need $\frac{100}{100}$ .) Simplify: $g cd (375, 1000) = 125$ , so $\frac{375 \div 125}{1000 \div 125} = \frac{3}{8} .$ Check the round-trip: $3 \div 8 = 0.375$ , and $0.375 \times 100 = 37.5%$ . ✓ We returned to the number we started from, so the conversion is sound. Answer: $\frac{a}{b} = \frac{3}{8}$ .

Recall Solution L5.2

Arrow used: the plum arrow (Decimal → Percentage, $\times 100$ ) at the end; the first step is plain algebra on the fraction. WHAT: $\frac{k}{40} = 0.225$ means $k = 0.225 \times 40$ . Compute: $0.225 \times 40 = 9$ . So $k = 9$ , i.e. the fraction is $\frac{9}{40}$ . To percent: $0.225 \times 100 = 22.5%$ . Sanity check: $9 \div 40 = 0.225$ ✓, and $\frac{9}{40}$ is a bit under a quarter ( $\frac{10}{40} = 0.25$ ), matching $22.5% < 25%$ . Answer: $k = 9$ , value $= \frac{9}{40} = 22.5%$ .

Recall Solution L5.3

Arrow used: the teal arrow (Percentage → Decimal, $\div 100$ ), then the teal arrow (Decimal → Fraction, place value + simplify). The master key $p % = \frac{p}{100}$ does not care whether $p$ is big or negative. (a) $250%$ : replace $%$ with $\div 100$ : $250% = \frac{250}{100} = 2.5 = \frac{5}{2}$ . WHY it can exceed one whole: "per hundred" does not cap at 100 — $250%$ is " $250$ out of $100$ ", i.e. two-and-a-half wholes. On the number line it sits past $1$ , at $2.5$ . Percentages over $100%$ are completely normal (e.g. a price that more than doubles). (b) $- 20%$ : same rule: $- 20% = \frac{- 20}{100} = - 0.2 = - \frac{1}{5}$ . WHY a minus survives untouched: dividing a negative by $100$ stays negative. A negative percentage just marks a change in the opposite direction (a fall rather than a rise). Its costume-swaps are ordinary: $- 0.2$ and $- \frac{1}{5}$ . Answer: (a) $250% = 2.5 = \frac{5}{2}$ ; (b) $- 20% = - 0.2 = - \frac{1}{5}$ .

Recall Solution L5.4

Arrow used: forward — orange arrow (Fraction → Decimal, divide) then plum arrow (Decimal → Percentage, $\times 100$ ); backward — teal arrow (Percentage → Decimal, $\div 100$ ) then teal arrow (Decimal → Fraction, place value + simplify). Forward. $7 \div 4 = 1.75$ (since $\frac{7}{4}$ is $1$ whole plus $\frac{3}{4} = 0.75$ ). Then $1.75 \times 100 = 175%$ . WHY over 100%: $\frac{7}{4}$ is bigger than one whole, so it must exceed $100%$ — a percentage over 100 is simply "more than one whole", nothing exotic. Backward (the point of this exercise). Take $175%$ and undo: $175% = \frac{175}{100} = 1.75$ . Now the teal place-value arrow: $1.75$ has its last digit two places after the point (hundredths), so $1.75 = \frac{175}{100}$ . Simplify with $g cd (175, 100) = 25$ : $\frac{175 \div 25}{100 \div 25} = \frac{7}{4} .$ We landed exactly back on $\frac{7}{4}$ — the ring closes even for values above $100%$ . Answer: $\frac{7}{4} = 1.75 = 175%$ , and $175%$ converts back to $\frac{7}{4}$ .

Recall Solution L5.5

Arrow used: the teal arrow (Percentage → Decimal, $\div 100$ ) turns each change into a multiplier; the plum arrow ( $\times 100$ ) puts the final answer back into percent. Convert the two changes to decimal multipliers:

" $+ 20%$ " survives-and-grows to $1 + 0.20 = 1.2 = \frac{6}{5}$ .
" $- 20%$ " keeps $1 - 0.20 = 0.8 = \frac{4}{5}$ . Chain them (order doesn't matter for multiplication): $\frac{6}{5} \times \frac{4}{5} = \frac{24}{25} = 0.96 = 96%.$ WHY not 100%: the second $20%$ is taken from the larger price, so it removes more than the first added. The final price is $96%$ of the original — not back to the start. Answer: final $= \frac{24}{25}$ of the original $= 96%$ (a $4%$ net loss).

Recall One-line self-test (answers hidden)

$\frac{7}{8}$ as a percent ::: $87.5%$ $0.625$ as a fraction ::: $\frac{5}{8}$ $\frac{2}{3}$ as an exact percent ::: $66 \frac{2}{3} %$ $0. \overline{45}$ as a fraction ::: $\frac{5}{11}$ $37.5%$ as a fraction ::: $\frac{3}{8}$ $\frac{7}{4}$ as a percent ::: $175%$ $250%$ as a decimal and fraction ::: $2.5 = \frac{5}{2}$ $- 20%$ as a decimal ::: $- 0.2$ $+ 20%$ then $- 20%$ leaves you at ::: $96%$ of the start

Connections

Converting - fractions ↔ decimals ↔ percentages (parent)
Fractions - simplifying and equivalent fractions
Decimal place value
Long division
Recurring decimals to fractions
Ratio and proportion
Percentage increase and decrease