Split a total fairly by counting parts, change units by multiplying by a clever "1," and learn to track the unit as carefully as the number.
Point 2 of 5 in this lesson: 4.6.2 Continued sharing: three-way mixes
All through this stage you have been comparing quantities with ratios and percents — a recipe is 2 : 3 flour to water, a class is 60% girls. Now we put ratios to work in two everyday jobs: splitting a total fairly and changing units. Both come down to the same humble move you have used since first grade — counting equal parts — but with the care to keep track of what each part is.
By the end of this lesson you will be able to: share any total in a given ratio by counting parts; handle three-way mixing problems like cement, sand, and gravel; convert measurement units by multiplying by a conversion ratio that is secretly a "1"; treat every quantity as a number × unit and know which ones you may add; and read a compound unit like km/h as two units divided. We keep one steady habit of color: the first quantity (the antecedent, the "part") is amber, the second quantity (the consequent, the "whole," the "out of") is blue, and a third term is purple. Increases stay green; decreases stay red.
Suppose you and a friend earn a $50 bonus together, but you did more of the work, so you agree to split it in the ratio 3 : 2 — three shares for you, two for them. How much does each of you get? The wrong instinct is to grab "$3 and $2" or to fumble with fractions. The right instinct is wonderfully simple: a ratio tells you how many equal parts there are.
Read 3 : 2 as three parts and two parts. Add them: 3 + 2 = 5 equal parts in total. The whole $50 is those 5 parts, so one part is
one part = total ÷ (total parts) = $50 ÷ 5 = $10.
Now just hand out parts. You get 3 parts: 3 × $10 = $30. Your friend gets 2 parts: 2 × $10 = $20. And the safety check that you should always do: the shares must add back to the total. $30 + $20 = $50. ✓
To split a total in the ratio a : b:
① Add the parts: total parts = a + b.
② Find one part: one part = total ÷ (total parts).
③ Hand them out: each share = (its part-count) × (one part).
Then check: the shares must add up to the total.
Share a 72 cm ribbon between two bows in the ratio 5 : 3.
① Parts: 5 + 3 = 8.
② One part: 72 ÷ 8 = 9 cm.
③ Shares: 5 × 9 = 45 cm and 3 × 9 = 27 cm.
Check: 45 + 27 = 72 cm. ✓
In 3 : 2, your share is not 32 of the money. It is 35 of it — three out of the five total parts. Always turn a ratio into "out of the total parts" before you read it as a fraction of the whole.
Set a total and a ratio a : b. The widget adds the parts, finds one part, and splits the bar into a amber + b blue pieces. The two shares always add back to the total — even when one part isn't a whole number.
The very same recipe works when a total is split three (or more) ways — which is exactly what happens in mixing problems. A standard concrete mix uses cement, sand, and gravel in the ratio 2 : 3 : 5. The colon now has two gaps, but nothing about the method changes: you still add up the parts and find what one part is worth.
Say you need 40 kg of concrete. Add the parts: 2 + 3 + 5 = 10 parts. One part = 40 ÷ 10 = 4 kg. Now hand them out:
cement: 2 × 4 = 8 kg · sand: 3 × 4 = 12 kg · gravel: 5 × 4 = 20 kg.
Check: 8 + 12 + 20 = 40 kg. ✓ The biggest part-count gets the most material — gravel here — which matches your intuition for what concrete is mostly made of.
A $120 prize is split among three friends in the ratio 1 : 2 : 3.
Parts: 1 + 2 + 3 = 6. One part: $120 ÷ 6 = $20.
Shares: $20, $40, $60. Check: 20 + 40 + 60 = $120. ✓
Set a total and a three-way ratio a : b : c. The bar fills with a amber, b blue, and c purple parts, and the three amounts always add back to the total.
Here is a secret that makes every unit conversion easy: a conversion fact is a ratio equal to 1. We know 1 m = 100 cm — they are the very same length. So if you divide one by the other, you must get 1:
100 cm1 m = 1 and 1 m100 cm = 1.
Both of those fractions are conversion factors, and both equal 1. Multiplying any quantity by 1 never changes its value — only how it looks. The trick is to pick the version of "1" that puts the old unit on the bottom, so it cancels, leaving the new unit standing.
Turn 2.5 m into centimeters. The old unit is m, so choose the factor with m on the bottom:
2.5 m × 100 cm1 m = 2.5 × 100 cm = 250 cm.
The m on top cancels the m on the bottom, exactly like cancelling a common factor in a fraction, and the answer comes out in cm. Going the other way, to shrink the number you divide. Turn 3000 g into kilograms, where 1 kg = 1000 g:
3000 g × 1 kg1000 g = 30001000 kg = 3 kg.
A conversion factor like 100 cm1 m equals 1, so it changes the units without changing the amount. Always write the factor with the old unit on the bottom so it cancels and the new unit survives.
Going to a smaller unit (m → cm), the number gets bigger (you need more of the small unit). Going to a bigger unit (g → kg), the number gets smaller. If your answer drifts the wrong way, you flipped the factor — turn it over and try again.
Pick a conversion and a value. The widget builds the correct factor (old unit on the bottom), cancels it, and shows the underlying 1 : factor ratio.
Every measurement is two things glued together: a number and a unit. "3 m" is the number 3 wearing the unit meters. The number alone — just "3" — is meaningless until you say three what. This habit of always carrying the unit is called dimensional sense, and it will quietly catch errors for the rest of your math and science life.
quantity = number × unit.
The wonderful part is that units follow the algebra too. When you multiply a quantity by a plain number (a "scalar"), the unit just comes along for the ride:
3 m × 2 = (3 × 2) m = 6 m.
But adding is fussier. You can only add quantities that share the same unit — just as you can add 3 apples + 2 apples but not 3 apples + 2 oranges. So 3 m + 2 m = 5 m is fine, but 3 m + 50 cm is a trap: the units don't match. You must first convert to a common unit, then add:
3 m + 50 cm = 300 cm + 50 cm = 350 cm (= 3.5 m).
A shelf needs 4 boards, each 1.2 m long. Total length?
4 × 1.2 m = (4 × 1.2) m = 4.8 m. The unit m rides through untouched — only the number is multiplied.
"3 m + 50 cm = 53" is wrong — you added a number of meters to a number of centimeters as if the units were the same. Adding (and subtracting) demands matching units. Multiplying and dividing don't, because there the units transform instead of needing to match (you'll see that next in km/h).
Build a quantity as number × unit, then choose an operation. Multiplying by a scalar keeps the unit; adding two quantities only works when the units match — otherwise the widget converts first, then adds.
When you divide one quantity by another, the units divide too — and that gives you a rate with a compound unit. Drive 150 km in 3 h, and your speed is
150 km3 h = 1503 km/h = 50 km/h.
Read the slash as "per": 50 km/h is "fifty kilometers per hour" — fifty kilometers for each one hour. The number 150 ÷ 3 = 50 is computed exactly as you'd expect, and separately the units divide: km ÷ h becomes the single compound unit km/h. The same shape produces every familiar rate:
distance ÷ time → km/h (speed) · mass ÷ volume → g/cm³ (density) · price ÷ weight → $/kg (unit price)
Unit price: a 4 kg bag of rice costs $12. Price per kilogram = $124 kg = $3/kg.
Density: a metal block has mass 60 g and volume 20 cm³. Density = 60 g20 cm³ = 3 g/cm³.
In both, the numbers divide and the two units stack into one compound unit.
"km/h" is a single unit of speed, just like "m" is a single unit of length. Don't read 50 km/h as "50 km and also some hours" — it means that in each one hour, 50 km are covered. To compare m/s with km/h you'd convert both the distance unit and the time unit, but the idea — one distance per one time — stays the same.
Enter a distance and a time. The widget divides the numbers and stacks the units into a compound unit — the speed in km/h. Switch to "general rate" to see price-per-kg or density form the same way.
To share a total in a ratio, add the parts, find one part (total ÷ total parts), and hand out part-counts — then check the shares sum to the total; this works for two parts or three or more. A unit conversion is multiplication by a conversion ratio that equals 1 (like 100 cm1 m), arranged so the old unit cancels. Every quantity is a number × unit; you may scale it by a number freely, but you may only add like units, converting first if they differ. And when you divide two quantities, the units divide into a single compound unit — km/h, $/kg, g/cm³ — a rate that says "so much per one of something."
You have now mastered ratios, proportion, and percentages. Next you cross to the other side of zero: Stage 5 — Negative & Rational Numbers. We'll extend the number line to the left, make sense of debts and temperatures below zero, and learn the rules for adding, subtracting, multiplying, and dividing signed numbers.
Work each one out first, then open the answer to check your thinking.
Six questions to lock it in. Tap the answer you think is right.
This lesson applies ratio reasoning to two practical jobs and is aligned to the U.S. Common Core ratio-and-proportion standards. Sharing a total in a ratio and the three-way mixing problems develop general ratio and rate problem solving (6.RP.A.3, extended in the grade-7 ratio cluster 7.RP.A). The unit-conversion section is squarely 6.RP.A.3d — convert measurement units by multiplying or dividing using ratio reasoning and manipulating units appropriately — taught here through the "conversion factor equals 1" idea and explicit factor-label cancellation. Sections 4.6.4 and 4.6.5 deliberately go beyond the minimum to build dimensional sense: a quantity is a number × unit, like units may be added while unlike units may not, and dividing quantities produces a compound-unit rate (km/h, $/kg, g/cm³). That habit of tracking units is the foundation students will lean on for rates and slope in later algebra and for every formula in middle- and high-school science. The most common misconception this lesson targets is treating a ratio term as a fraction of the whole (it is a fraction of the total parts) and adding unlike units without converting.