1.1.19 · D5Arithmetic & Number Systems
Question bank — Ratio and proportion — equivalent ratios, dividing in a ratio
True or false — justify
Every answer must give the reason, not just T/F.
and are equivalent because both go up by 1 on the left and stay comparable.
False. Equivalence needs the same fraction: but . Adding to terms changes the value; only multiplying both by one non-zero number preserves it.
A ratio always has no units.
True. Both terms are the same kind of quantity, so the units cancel in , leaving a pure comparison number.
The ratio is .
False. You must convert to the same unit first: . Comparing across different units gives a meaningless number.
If (all terms positive) then is also equal to .
True, with conditions. Since we get — provided (guaranteed if terms are positive). Picture two identical staircases pushed together: the steepness is unchanged. This is adding corresponding parts of equal ratios, totally different from adding a number to both terms of one ratio.
is the same statement as .
False. Order encodes who is being compared to whom; . Swapping terms inverts the fraction.
Multiplying both terms of a ratio by gives an equivalent ratio.
False. You get , whose fraction is undefined. The scaling factor must be non-zero.
Scaling by gives an equivalent ratio .
True as a fraction (), but the negative flips the direction/order of both parts. In real sharing problems, negative parts have no meaning — a negative scale factor reverses sign, so treat it as a warning, not a valid split.
Simplifying to loses information about the original quantities.
True in one sense. tells you the relative sizes but not that the originals were and ; the ratio deliberately throws away absolute size and keeps only the pattern.
In the proportion , the terms and are the means.
False. and are the extremes (outer terms); and are the means (inner terms). Cross-multiplication says extremes-product means-product.
Spot the error
Each line hides one wrong step. Name it.
"To divide in , each part is ."
The error is dividing by a single term. You divide by the sum of parts, , giving one part ; then .
" equals because I subtracted 2 from the first and 3 from the second."
Wrong operation. It happens to land on by luck, but the correct rule is to divide both by . Subtraction fails in general — try it on .
" so ."
The products are paired wrong. Cross-multiplication (valid because ) gives extremes extremes vs means means: , not .
"Split in : one part since there are three people."
Confused number-of-people with number-of-parts. Total parts , so one part ; shares are .
"P and Q share in ; P is bigger since 3 comes first."
Position doesn't mean size. The larger number gets the larger share: Q with parts gets more. Order tells you the pairing, not the ranking.
"To make into an equivalent ratio with first term , I add 4 to get ."
Adding again. To turn into you multiply by , so you must multiply both: .
" can't be a ratio because it has decimals."
Nothing forbids decimal terms; scale both by to get . A ratio just needs both terms to be the same kind of number.
Why questions
Why must the two quantities in a ratio be of the same kind?
So their units cancel and becomes a pure scaling number. "" cannot be reduced to one meaningful comparison.
Why does multiplying both terms by the same leave the ratio unchanged?
Because ; the factor cancels top and bottom. Picture it: photocopying a "2 tall, 3 wide" box at any zoom keeps its shape — the tall-to-wide feeling is fixed even as the box grows.
Why do we divide by the to get simplest form, not just any common factor?
The gcd is the largest common factor, so after dividing by it no further common factor remains — the ratio can't be reduced again.
Why does cross-multiplication turn a proportion into something easier?
Imagine two fractions as two see-saws that must balance. Multiplying both sides by (allowed since ) lifts the numbers off the division lines, leaving the clean balance — no fractions to juggle. Visually, you're re-drawing both slopes as areas of two rectangles that must be equal.
Why do we divide the total by the sum of the parts?
Look at the strip figure: the whole bar is chopped into equal blocks. One block's value is therefore , and each person collects their number of blocks. Dividing by the sum literally measures one block.
Why can a ratio be seen as a fraction but a fraction isn't always thought of as a ratio?
A ratio is the fraction by definition (see Fractions and simplification), but "fraction" often means a part of one whole, while "ratio" emphasises comparison between two separate quantities. Same maths, different framing.
Why does the "parts" method extend to three or more terms with no new rule?
Each term still counts equal blocks in the same bar; you simply add all terms for the total blocks. is one bar cut into blocks, handed out — the picture never changed, just gained a third recipient.
Why is a percentage just a special ratio?
A percentage is a ratio whose second term is fixed at (see Percentages); "40%" means , comparing to a hundred every time.
Edge cases
Is the ratio allowed?
No. Since means and division by zero is undefined, is simply not a valid ratio. You cannot simplify it, form a proportion with it, or divide a total in it.
Dividing in ratio — what does each get?
One part , so both shares are . Here the sum of parts is , so the division is legal; there's simply nothing to split.
What does the ratio (equal terms, ) represent?
It simplifies to — the two quantities are equal in size. Dividing a total in means splitting it in half.
If one term of a ratio is much larger, can the ratio still be ?
No. If the terms are equal after simplifying, the quantities must be equal; a large absolute gap means a non- ratio. Ratio depends on relative, not absolute, size.
In , what happens if ?
The left ratio is undefined (division by zero), so the proportion has no meaning. Proportions require all "denominator" terms — here and — to be non-zero.
Can a ratio have a negative term?
Formally the fraction still exists (e.g. ), but in counting/sharing contexts terms are taken positive. A single negative sign flips the order of comparison, so mixing signs is a classic trap — always check the sign before declaring two ratios "equal".
Can equivalent ratios have terms that aren't whole numbers?
Yes. , , and are all equivalent; scaling by any non-zero number — including a fraction — preserves the ratio, as with Similar figures whose sides scale by any factor.
Is in simplest form?
No. All three terms share , so divide through to get . For multi-term ratios, use the gcd of all terms.
Connections
- Ratio and proportion — equivalent ratios, dividing in a ratio — the parent these traps stress-test.
- Fractions and simplification — why behaves like .
- Highest Common Factor (GCD) — the reason gcd gives simplest form.
- Percentages — the ratio-out-of-100 special case.
- Unitary method — "one part" is the unitary idea in disguise.
- Direct and inverse proportion — proportions that stay constant.
- Similar figures — equal side-ratios with non-integer scale factors.