How this instrument works
An equivalent fraction is simply another name for the same rational number. Two thirds, four sixths, and eight twelfths look like three different fractions, but each divides the number line at the identical point — one value wearing three different sets of digits. This instrument produces new members of that family by multiplying both the numerator and the denominator of a starting fraction by the same scale factor k.
The reason multiplication preserves the value while other operations would ruin it comes down to what k⁄k actually equals: exactly 1, for any nonzero k. Multiplying a fraction by k⁄k multiplies it by 1, so nothing about its size changes, only how finely the whole is sliced. Picture a pizza cut into 3 equal slices with 2 shaded: cut every one of those slices into 4 smaller pieces instead, and 8 of the resulting 12 pieces are shaded. Four times as many cuts, four times as many shaded pieces, and exactly the same amount of pizza.
The mistake this trips up most often is reaching for addition instead of multiplication: adding 1 to both the top and bottom of 2⁄3 gives 3⁄4, a fraction that is not equivalent at all — 2⁄3 is about 0.667 and 3⁄4 is 0.75. Scaling has to be a single multiplicative factor applied to both terms together, never a sum tacked onto each. Running the scale factor below 1 works in reverse, shrinking a fraction toward its simplest form, and a negative k carries both terms' signs along together, so −2⁄−3 still lands on the same 0.667 as the fraction it started from.
- Enter the fraction you're starting from into Numerator and Denominator — 2 and 3, say, for two thirds.
- Set Scale factor to the multiplier you want applied to both terms at once — 4 turns two thirds into eighths and twelfths.
- Read Equivalent numerator and Equivalent denominator together as the new fraction; the ratio between them still equals the original.
- Try a Scale factor between 0 and 1, such as 0.5, to watch the fraction shrink toward a smaller equivalent form instead of growing.
- A negative Scale factor flips the sign of both results at once, which leaves the fraction's actual value unchanged.
Worked example — two thirds scaled by 4
Enter Numerator = 2, Denominator = 3, and Scale factor = 4. Equivalent numerator returns 8.0, since 2 × 4 = 8, and Equivalent denominator returns 12.0, since 3 × 4 = 12 — the pair 8⁄12 is the fraction 2⁄3 written in twelfths instead of thirds.
Nothing about the underlying value moved: 2 ÷ 3 and 8 ÷ 12 both equal 0.6666666666666666 to machine precision, and cross-multiplying confirms it algebraically too, since 2 × 12 = 24 and 3 × 8 = 24 — the two products matching is exactly the textbook test for two fractions being equivalent.
Questions
What makes two fractions equivalent?
Two fractions are equivalent when they mark the identical point on the number line, even though their numerators and denominators differ. Every equivalent fraction is generated from another by multiplying both numerator and denominator by the same nonzero number k — 2⁄3 and 8⁄12 are equivalent because 8 = 2 × 4 and 12 = 3 × 4, the same k applied to both terms.
Why can't I just add the same number to the top and bottom instead?
Because addition and multiplication change a ratio in completely different ways. Adding 1 to both terms of 2⁄3 gives 3⁄4, but 2⁄3 ≈ 0.667 while 3⁄4 = 0.75 — not the same value at all. Only a multiplicative scale factor, applied to numerator and denominator together, leaves the ratio between them, and therefore the fraction's value, unchanged.
How do I reduce a fraction to its simplest equivalent form?
Run the scaling in reverse with a fractional k: divide both numerator and denominator by their greatest common divisor. 8⁄12 has a greatest common divisor of 4, so dividing both terms by 4 — the same as scaling by k = 1⁄4 — returns 2⁄3, the simplest fraction in that entire equivalent family.
Does the scale factor have to be a whole number?
No — any nonzero number works, including decimals. Scaling 5⁄8 by k = 0.5 gives 2.5⁄4, which still equals 5⁄8 exactly (0.625 both ways), because the defining property only requires the same k applied to both terms, not that k be an integer.
How can I check whether two fractions I already have are actually equivalent?
Cross-multiply: for a⁄b and c⁄d, the fractions are equivalent exactly when a × d equals b × c. For 2⁄3 and 8⁄12, that's 2 × 12 = 24 against 3 × 8 = 24 — matching products confirm the two fractions represent the same value without needing to divide either one out to a decimal first.
What happens if the scale factor is negative?
Both terms flip sign together, and the fraction's value stays exactly the same, because a negative divided by a negative is positive again. Scaling 2⁄3 by k = −4 gives −8⁄−12, which still equals 0.6666666666666666 — the identical value as the original, just written with two negative terms instead of two positive ones.