Secondary 1 Fractions and Their Notations Worksheet

15 of the 16 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 15 of the 16 questions are printed below. The other 1 is built on a diagram or a table of values that does not translate to the page, so it is in the free PDF — marked below where it would have come.

1. Q1Adding Fractions

   A snow-clearing driver works through one route over three days. On Monday he clears $\frac{3}{8}$ of the route and on Tuesday he clears $\frac{1}{6}$ of it.

   1. What fraction of the route has he cleared after Tuesday?
   2. On Wednesday he clears another $\frac{5}{24}$ of the route. What fraction of the route is cleared in total? Write your answer in lowest terms.
   3. Compute $\frac{2}{5}+\frac{3}{4}$ and write the result as a mixed number.
2. Q2Dividing Fractions

   A bubble-tea counter serves tapioca pearls with a $\frac{3}{4}$-cup scoop.

   1. How many full scoops can be served from $4\frac{1}{2}$ cups of pearls?
   2. A $\frac{5}{6}$ kg bag of sugar is shared equally among $10$ jars. How many kilograms go into each jar?
3. Q3Equivalent Fractions and Simplifying Fractions
   1. Write ${\displaystyle \frac{42}{56}}$ in lowest terms.
   2. Write ${\displaystyle \frac{90}{144}}$ in lowest terms.
   3. Complete: ${\displaystyle \frac{7}{12}}={\displaystyle \frac{?}{84}}$.
   4. Are ${\displaystyle \frac{36}{60}}$ and ${\displaystyle \frac{3}{5}}$ equivalent? Justify.
4. Q4Expressing a Decimal Number as a Percentage and Vice Versa
   1. Write $0.375$ as a percentage.
   2. Write $1.4$ as a percentage.
   3. Write $6\%$ as a decimal number.
   4. Write $12.5\%$ as a decimal number.
5. Q5Expressing a Fraction as a Decimal Number and Vice Versa
   1. Write ${\displaystyle \frac{7}{8}}$ as a decimal number.
   2. Write ${\displaystyle \frac{9}{20}}$ as a decimal number.
   3. Write $0.64$ as a fraction in lowest terms.
   4. Write $0.35$ as a fraction in lowest terms.
6. Q6Expressing a Fraction as a Mixed Number and Vice Versa
   1. Write ${\displaystyle \frac{47}{6}}$ as a mixed number.
   2. Write $5\frac{3}{8}$ as an improper fraction.
   3. Write ${\displaystyle \frac{100}{12}}$ as a mixed number in lowest terms.
7. Q7Expressing a Fraction as a Percentage and Vice Versa
   1. Write ${\displaystyle \frac{3}{8}}$ as a percentage.
   2. Write ${\displaystyle \frac{9}{25}}$ as a percentage.
   3. Write $65\%$ as a fraction in lowest terms.
   4. Write $12.5\%$ as a fraction in lowest terms.
8. Q8Expressing a Fraction as a Periodic Number and Vice Versa
   1. Write ${\displaystyle \frac{4}{9}}$ as a periodic decimal number, using the bar notation.
   2. Write ${\displaystyle \frac{7}{11}}$ as a periodic decimal number.
   3. Write ${\displaystyle \frac{5}{12}}$ as a periodic decimal number. Careful: the repeating part does not begin right after the decimal point.
   4. Knowing that ${\displaystyle \frac{1}{9}}=0.\bar{1}$, write $0.\bar{8}$ as a fraction.
9. Q9Expressing a Mixed Number as a Decimal Number and Vice Versa
   1. Write $3\frac{3}{4}$ as a decimal number.
   2. Write $2\frac{1}{8}$ as a decimal number.
   3. Write $5.6$ as a mixed number in lowest terms.
   4. Write $7.24$ as a mixed number in lowest terms.
10. Q10Fractional Notation (Fractions)
    1. In the fraction ${\displaystyle \frac{7}{12}}$, name the numerator and the denominator, and say what each one tells you.
    2. In a class of $32$ students, ${\displaystyle \frac{5}{8}}$ chose the science museum for the class trip. How many students is that?
    3. Three identical granola bars are shared equally among $4$ campers. Write one camper's share as a fraction, and explain what this shows about the link between a fraction and a division.
11. Q11Multiplying Fractions
    1. Compute ${\displaystyle \frac{2}{3}}\times {\displaystyle \frac{9}{10}}$ and write the result in lowest terms.
    2. A video game gives a player $48$ inventory slots. Two thirds of them are unlocked, and ${\displaystyle \frac{3}{4}}$ of the unlocked slots are already full. How many slots are full?
    3. Compute $1\frac{1}{5}\times {\displaystyle \frac{5}{9}}$.
12. Q12Ordering Fractions and Mixed Numbers
    1. Put ${\displaystyle \frac{3}{4}}$, ${\displaystyle \frac{5}{6}}$, ${\displaystyle \frac{7}{8}}$ and ${\displaystyle \frac{11}{12}}$ in order from smallest to largest.
    2. Put $2\frac{1}{4}$, ${\displaystyle \frac{9}{4}}$, $2\frac{2}{5}$ and ${\displaystyle \frac{11}{5}}$ in order from smallest to largest, using $<$ and $=$ where each belongs.
13. Q13Subtracting Fractions
    1. Compute ${\displaystyle \frac{7}{8}}-{\displaystyle \frac{1}{3}}$.
    2. Compute $5\frac{1}{4}-2\frac{2}{3}$ and write the result as a mixed number.
    3. A rainwater barrel is ${\displaystyle \frac{5}{6}}$ full. A gardener uses an amount equal to ${\displaystyle \frac{3}{8}}$ of the full barrel. What fraction of the barrel is left?
14. Q14The Common Denominator — Secondary 1, 2 and 3
    1. Rewrite ${\displaystyle \frac{5}{12}}$ and ${\displaystyle \frac{7}{18}}$ with their least common denominator, then say which of the two is larger.
    2. Rewrite ${\displaystyle \frac{3}{8}}$, ${\displaystyle \frac{5}{6}}$ and ${\displaystyle \frac{7}{12}}$ with their least common denominator.
15. Q15Types of Fractions

    For each number below, say whether it is a proper fraction, an improper fraction or a mixed number. Then, looking only at the five that are written as fractions, state which of those are already in lowest terms, and which one is a unit fraction. $$\frac{5}{9},\frac{11}{4},3\frac{1}{2},\frac{6}{6},\frac{8}{12},\frac{1}{7}$$

16. Q16Synthesis — drawing on several sheets in this topic

    This question is built around a diagram or a table of values. Open it in the PDF.

The 16 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

The denominator names the unit

A fraction is not one number; it is a count and a unit. The denominator says what has been counted — twelfths, twenty-fourths, fifths — and the numerator says how many of them there are. Nearly every rule on this topic falls out of that one sentence, including the one people find most arbitrary: you cannot add 3/8 and 1/6 as they stand for the same reason you cannot add three metres and one inch, and the fix is the same, which is to put both into the same unit first.

The fraction bar is a division sign (Q10)

Q10 ends with three granola bars shared among four campers, and the point of it is the equality 3/4 = 3 ÷ 4. Cut every bar into quarters and there are twelve quarters to hand out, three each. That identity is what makes every other conversion on this sheet possible: 7/8 becomes 0.875 because 7 ÷ 8 = 0.875, and there is nothing more to the rule than that.

The same question opens with the meaning of the two numbers, which is worth answering in full sentences even though it looks like the easiest part of the sheet. "The denominator says the whole was cut into twelve equal parts; the numerator says seven of them are taken" is the sentence that later explains why unequal parts cannot be counted as a fraction at all.

Every conversion, in one place (Q4 to Q9)

Six questions here are the same skill pointed in different directions, and it is worth seeing them as one system rather than six rules to memorise.

Two of them carry a warning. A percentage above 100 is not an error — Q4 asks for 1.4 as a percentage, and 140% simply means more than one whole. And in Q7, 12.5% converts cleanly to 1/8, which is a useful reminder that a percentage is allowed a decimal in it.

Decimals that never stop (Q8)

Some divisions never terminate, and the bar notation is how you write the result honestly rather than rounding it away. 4/9 is 0.4 repeating; 7/11 repeats a two-digit block, 0.63; and 5/12 is the one that catches people, because the repetition does not start immediately — it is 0.41 with only the 6 repeating afterwards. Put the bar over exactly the digits that recur and no others.

Going the other way, the ninths are the family to know. Since 1/9 is 0.1 repeating, 0.8 repeating is eight times that, which is 8/9. Anchoring on a known fraction and scaling is much safer at this level than trying to reconstruct the fraction from the decimal.

Least common denominator, and why "least" is a convenience (Q14)

Any common multiple of the denominators will do — multiplying them together always works. The least one is preferred because it keeps the numbers small and usually leaves less simplifying at the end, which is where errors get made. Q14 builds it properly: factor each denominator, take each prime the greatest number of times it appears, multiply. For 12 and 18 that gives 36, not 216.

Its second part does three denominators at once, and the method does not change — which is the point of including it.

Ordering: pick one notation and put everything into it (Q12)

Q12's first part is four fractions with denominators 4, 6, 8 and 12, and a common denominator of 24 turns them into four numerators you can read off in order. Its second part mixes mixed numbers with improper fractions, and decimals are the faster route there — with a catch built in on purpose, because two of the four values are the same number written two ways, so the answer needs an equals sign in it as well as the inequality signs.

Mixed numbers, and the subtraction that needs regrouping (Q6, Q9, Q13, Q15)

Q15 sorts the vocabulary out: proper when the numerator is smaller than the denominator, improper when it is not, mixed when a whole number sits beside a proper fraction. Notice that 6/6 counts as improper — the test is "numerator not smaller", not "numerator bigger" — and that a mixed number is not a fraction, which matters when a question asks you to look only at the ones written as fractions.

Q13's second part is the classic. Subtracting mixed numbers, the fractional parts may not cooperate, and the temptation is to subtract the smaller fifths from the larger and carry on. Two safe routes: convert both to improper fractions and subtract, or regroup one whole from the whole-number part into fifths — writing 4 and 1/5 as 3 and 6/5 — and then subtract in place. Check by adding the answer back to what you subtracted.

Q16: the synthesis question

A class vote with the three shares recorded in three different notations — one fraction, one percentage, one decimal — which is the situation the whole topic exists for. Put all three into the same notation, check they account for the whole class, then turn each share into a number of students. The final check is that the three counts add back to the size of the class. (This question is laid out as a table, so it lives on the printed sheet rather than on this page.)

Getting the most out of it

Decide the notation before you start

Look at what the question is actually asking you to do, then choose the costume that suits it: percentages for comparing shares, decimals for ordering a mixed list, fractions for exact arithmetic. Converting first and computing second is almost always quicker than converting halfway through.

Estimate the answer before you compute it

Adding a positive amount must make the total bigger; subtracting must make it smaller; multiplying by something less than one must shrink a number and by something more than one must grow it. One of the questions here can be marked wrong on that basis alone, with no arithmetic at all — and that is a check you can run on your own work every time.

Finish in lowest terms, and check by going backwards

Divide both terms by their GCF at the end, and read the question again to see which notation the answer was wanted in. Then verify: add the answer back after a subtraction, multiply back after a division, convert back after a conversion. Every question on this sheet is short enough that the check costs a line.

Common questions