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Fractions

3/8 as a decimal, 7/20 as a percent, 0.375 as a fraction, 18/24 in lowest terms, 5/8 against 3/5 — exact conversions with denominators that keep growing.

  • Number sense
  • Timed
  • Endless levels

How it plays

Five kinds of question: a fraction as a decimal, a fraction as a percent, a decimal as a fraction in lowest terms, a fraction reduced to lowest terms, and which of two fractions is larger.

A fraction ends as a decimal when its bottom is made only of 2s and 5s: multiply top and bottom up to a power of ten (3/8 = 375/1000). To compare two fractions, cross-multiply. Levels grow the denominators without end and shorten the time.

The wrong options are the usual slips: the digits run together (3/8 read as 0.38), the decimal point one place out, the fraction upside down, lowest terms not reached.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This trains moving between fractions, decimals and percents and judging their size. It is practice at this, nothing wider.

Learn

How to get better at fractions: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Name the kind of question first: to a decimal, to a per cent, back to a fraction, lowest terms, or which is larger. Each has its own one move.
  2. Look at the bottom number. If it is made of 2s and 5s, think “what makes it 10, 100 or 1,000?”. If it shares a factor with the top, think “what divides both?”.
  3. Do that one move: multiply top and bottom up to a power of ten, or divide both by their highest common factor, or cross-multiply.
  4. Check the size before you answer: 3/8 is less than ½, so its decimal must be below 0.5, and its per cent below 50%.

Techniques and tricks

  • Make the bottom a power of ten

    Every 2 in the bottom needs a 5 to make a 10, and every 5 needs a 2. Multiply top and bottom by whatever completes the pairs; the top is then the decimal’s digits.

    e.g. 7/25 = 28/100 = 0.28; 3/16 = 1,875/10,000 = 0.1875; 137/250 = 548/1,000 = 0.548

  • Know the eighths by heart

    1/8 is 0.125, and the rest step by 0.125: 3/8 = 0.375, 5/8 = 0.625, 7/8 = 0.875. Sixteenths are half as big: 1/16 = 0.0625.

    e.g. 5/16 = 0.3125, half of 5/8 = 0.625

  • A per cent is two places

    Find the decimal, then move the point two places right. Or make the bottom 100 directly when it divides 100.

    e.g. 7/20 = 35/100 = 35%; 3/8 = 0.375 = 37.5%

  • Back to a fraction: count the places

    Put the digits over 10, 100 or 1,000 — one zero per decimal place — then cancel. Because the bottom is only 2s and 5s, the only factors to cancel are 2s and 5s.

    e.g. 0.375 = 375/1,000. 375 = 3 × 125 and 1,000 = 8 × 125, so 3/8.

  • The common factor divides the difference

    Any number that divides both top and bottom also divides their difference. The difference is often the highest common factor itself, or close to it.

    e.g. 18/24: the difference is 6, and 6 divides both, so 18/24 = 3/4

  • Cross-multiply to compare

    a/b against c/d: compare a × d with c × b. The bigger product sits with the bigger fraction.

    e.g. 5/8 or 3/5: 5 × 5 = 25 against 3 × 8 = 24, so 5/8 is larger — by a hair

  • Benchmarks before arithmetic

    Place each fraction against ½ and 1 first. One above ½ and one below needs no working. Two just under 1 are compared by what they are missing.

    e.g. 7/8 or 8/9: they miss 1/8 and 1/9, and 1/9 is smaller, so 8/9 is larger

As the levels climb

  • Every kind is open from the first level; what climbs is the size of the numbers. The bottoms that end gain more 2s and 5s, so decimals grow to three, four and more places. Complete the 2–5 pairs one at a time rather than guessing the multiplier.
  • For lowest terms the common factor grows too — 2, 3 and 4 at first, then 6, 8, 9, 12 and 15. Divide by the highest common factor in one go, or keep dividing until no 2, 3 or 5 is left.
  • Comparisons move to bigger, closer fractions, like 46/53 against 10/33 or 2/3 against 91/94. Try a benchmark first; cross-multiply only when both sit on the same side of ½.
  • Between the size steps the clock shrinks, down to 62% of its start. Learn the eighths, sixteenths and twenty-fifths well enough that they cost no time.

Traps to avoid

  • The digits run together: 3/8 read as 0.38, or 7/20 as 0.720. A fraction’s decimal comes from dividing, never from writing the numbers side by side.
  • The decimal point one place out: 0.06 or 6 for 3/5. Check the size: 3/5 is more than ½, so it is 0.6.
  • Lowest terms not reached: 80/276 for 160/552, or 2/4 for 4/8. After dividing, ask again whether top and bottom still share a 2, 3 or 5.
  • The fraction upside down: 8/3 offered for 3/8, or 1.6 for 3/5. A proper fraction is less than 1, so its decimal starts with 0.
  • The bigger bottom taken for the bigger fraction. 5/9 has the bigger numbers, but 3/4 is larger: 3 × 9 = 27 beats 5 × 4 = 20.

How to practise

  • Use “Questions” to drill one kind at a time — “Fraction to decimal” on its own until the eighths are automatic, then “Decimal to fraction” to run the same facts backwards.
  • Switch “Answer by” to “Typing the answer” once you choose well. Fractions are typed as a/b, so you must reach lowest terms yourself rather than spot it.
  • Turn “Options shown” up to 6 or 8 to meet every classic slip in one question. “Which is larger” always shows its two fractions, whatever you set.

Levels

Each question has a clock that comes from the work it asks for. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

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