A ratio is not a pair of numbers — it is a shape of a comparison. The ratio 3:2 for orange juice to water does not say "there are three litres of juice and two litres of water." It says "for every three units of juice, there are two units of water," no matter what the total size turns out to be. The picture below lets you feel that claim directly. Drag the total volume and the two liquids grow or shrink together, always in the same proportion.

The picture

In a 3:2 mixture, the juice fills \tfrac{3}{5} of the beaker and the water fills \tfrac{2}{5} — because the total is 3+2 = 5 parts, and each liquid claims its share of those parts. The denominator of the share is the sum of the ratio terms, not either term on its own. This is the rule that trips up most students on their first ratio problem, and the beaker picture makes it visible.

Interactive beaker filled with orange juice and water in a three to two ratioA vertical beaker. Orange juice fills the bottom three fifths of the liquid; water fills the top two fifths. A slider below controls the total volume in millilitres from zero to one thousand. As the reader drags the slider, both liquids grow together, keeping the three-to-two proportion constant. Readouts show the current total volume, the juice volume (three-fifths of total), and the water volume (two-fifths of total). juice (3 parts) water (2 parts) 1000 mL 0 mL ↔ drag to change total volume
Drag the red slider to change the total volume. Both liquids scale together, keeping the juice-to-water ratio locked at three to two. At five hundred millilitres total, the beaker holds three hundred millilitres of juice and two hundred millilitres of water. At one litre, six hundred and four hundred. The shape of the comparison — "three to two" — stays the same; only the absolute quantities grow.

Why both grow together: the rule "juice to water is 3:2" is a constraint on the proportion, not the volume. If the proportion is fixed, doubling the total doubles each part. The mathematical statement is \tfrac{\text{juice}}{\text{water}} = \tfrac{3}{2}, which is a single equation in two unknowns — choose any total and the equation fixes both liquids.

Reading three numbers that move together

The three readouts at the top track the same underlying fact from three angles. At total = 500 mL, the juice reads 300 mL and the water reads 200 mL. Notice the checks.

These three relationships are different surfaces of the same fact. The beaker picture is a way of drawing a single equation, 3:2, and watching it stay true under rescaling.

Two volumes to drag to

At V = 250 mL. Juice is \tfrac{3}{5} \times 250 = 150 mL. Water is \tfrac{2}{5} \times 250 = 100 mL. Check: 150 + 100 = 250, and 150 : 100 = 3:2.

At V = 1000 mL (one litre). Juice is 600 mL. Water is 400 mL. Check: 600 + 400 = 1000, and 600 : 400 = 3:2 after dividing both sides by 200.

Notice that across these two settings, the juice volume went from 150 to 600 — a factor of 4. The water volume went from 100 to 400 — also a factor of 4. Both liquids scale by the same factor, because the ratio is fixed. This is called proportional scaling, and it is the single most important property of a ratio.

The unitary method, made visible

If a problem tells you "juice and water are mixed in the ratio 3:2 and the total is 750 mL; how much of each liquid is there?" — the beaker picture is the calculation.

Step 1. Add the ratio terms: 3 + 2 = 5. The total is being split into five equal "parts."

Step 2. Find one part: \tfrac{750}{5} = 150 mL. Each "part" is 150 mL.

Step 3. Scale up: juice = 3 \times 150 = 450 mL, water = 2 \times 150 = 300 mL.

Check: 450 + 300 = 750 \,\,\checkmark, and 450 : 300 = 3:2 after dividing by 150 \,\,\checkmark.

This is the unitary method — find what one unit is worth, then scale up — and the beaker picture is its visual translation. Each "part" is a fixed slice of the beaker, and each liquid simply claims its count of those slices.

The common trap: "juice is \tfrac{3}{2} of the water"

People often say, on first meeting a 3:2 ratio, "so juice is \tfrac{3}{2} of the total." That is wrong. Juice is \tfrac{3}{2} of the water, but it is \tfrac{3}{5} of the total. The beaker picture makes this obvious — the juice fills more of the beaker than the water does, but it clearly does not fill \tfrac{3}{2} = 1.5 times the whole beaker, because nothing can fill more than the whole beaker.

The shareholder fraction — the fraction of the total — is always

\frac{\text{your ratio term}}{\text{sum of all ratio terms}}.

For juice in a 3:2 mixture, that is \tfrac{3}{3+2} = \tfrac{3}{5}, not \tfrac{3}{2}.

Extending to three liquids

Ratios work the same way with three or more terms. A 3:2:1 mixture of juice, water, and lemon would split the beaker into 3+2+1 = 6 parts, with juice at \tfrac{3}{6} = \tfrac{1}{2}, water at \tfrac{2}{6} = \tfrac{1}{3}, and lemon at \tfrac{1}{6}. Check: \tfrac{1}{2} + \tfrac{1}{3} + \tfrac{1}{6} = \tfrac{3}{6} + \tfrac{2}{6} + \tfrac{1}{6} = 1. The fractions always sum to one, because they are shares of the whole.

Every ratio problem you will ever meet in school arithmetic is a version of this beaker — a total being sliced into parts, with each share's fraction being "its ratio term over the sum of ratio terms."

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