Ratio: 2 : 3 · 跟马丁学编程

A ratio is a recipe for mixing and sharing. You will learn to read it, simplify it and expand it.

A ratio is a recipe written with a colon

Juice is diluted with water in the ratio 1 : 4. Concrete is mixed from cement and sand 1 : 3. A map has a scale 1 : 50 000. What does that colon mean?

A ratio says how many parts of what stand next to each other. A juice ratio 1 : 4 means: for 1 part of syrup, 4 parts of water. A part can be a glass, a cup or a bucket — the ratio does not set the size of a part, only their number compared with each other.

That is why a ratio is a recipe: it works for a small glass and for a whole jug, as long as you keep the parts.

It is read in order: the first number belongs to the first thing. “Syrup and water 1 : 4” is not the same as 4 : 1 — that would be syrupy syrup with a drop of water.

And because a ratio is really a pair of numbers, it can be simplified and expanded like a fraction: 2 : 4 is the same ratio as 1 : 2. Today you will learn to read, write and tidy ratios. Sharing by them we will do next time.

Sofie mixes lemonade for a party: syrup and water 1 : 6. Adam brings a giant jug and pours by eye. “Stop,” Sofie halts him, “take a cup. One cup of syrup, six cups of water.” Adam: “And if the jug is bigger?” — “Then two cups of syrup and twelve of water. The ratio 2 : 12 is still 1 : 6, I just doubled the recipe.” Adam tastes and admits: “Last year I mixed 1 : 2 and nobody finished it.” Sofie laughs: “Because 1 : 2 is not lemonade, it is syrup with a hint of water.”

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Always read a ratio with the things, not just the numbers: “syrup to water 1 : 6”. Who only says “one to six” soon forgets what was first.

Reading, simplifying and expanding a ratio

Writing and reading. A ratio a : b we read “a to b”. Order is binding: a ratio of boys to girls 3 : 2 says that for 3 boys there are 2 girls.

Simplifying. You can divide both numbers of a ratio by the same number and the ratio does not change: 6 : 9 = 2 : 3 (divided by three). A ratio in simplest form has the smallest whole numbers — like a simplified fraction.

Expanding. You can multiply both numbers by the same number: 1 : 6 = 2 : 12. That is exactly how you enlarge a recipe.

A ratio and a fraction of the whole. The ratio 2 : 3 splits the whole into 2 + 3 = 5 parts. The first gets 2/5 of the whole, the second 3/5. You will need this link next time when sharing amounts.

A ratio from given values. In the class there are 12 boys and 18 girls → ratio 12 : 18, simplified 2 : 3.

Same units! A ratio 50 cm to 2 m cannot be written 50 : 2. Convert first: 50 cm : 200 cm = 1 : 4. A ratio compares numbers, and they must measure in the same unit.

A ratio can have three members too: cement, sand and gravel 1 : 2 : 3. The simplifying rules work the same.

Simplify the ratio 24 : 36 to simplest form

Step 1: I look for a common divisor of 24 and 36. Both can be divided by twelve.

Step 2: 24 : 12 = 2 and 36 : 12 = 3.

Result: 24 : 36 = 2 : 3. Who does not see twelve at once, simplifies step by step: divide by 2 → 12 : 18, divide by 2 → 6 : 9, divide by 3 → 2 : 3. Same result, just more steps. You know simplest form when the numbers have no common divisor left.

Recipe 1 : 6 for 3 cups of syrup

Step 1: ratio syrup : water = 1 : 6. I have 3 cups of syrup, so the first number has tripled.

Step 2: I expand the whole ratio by three: 1 : 6 = 3 : 18.

Result: for 3 cups of syrup belong 18 cups of water. In total you get 3 + 18 = 21 cups of lemonade. You always expand the whole ratio by the same number — who adds only syrup changes the taste, not the size of the batch.

Write as a ratio: a segment 40 cm and a segment 1 m

Step 1: the units differ — I convert to centimetres: 1 m = 100 cm.

Step 2: ratio: 40 : 100.

Step 3: I simplify by twenty: 40 : 100 = 2 : 5.

Result: the segments are in the ratio 2 : 5. Without converting units it would come out 40 : 1, which is nonsense — the first segment is definitely not forty times longer. Always match the units, only then write the colon.

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A ratio does not change by adding, only by multiplying and dividing. The ratio 1 : 2 is not the same as 2 : 3, even if you “only added one” to both numbers. You enlarge a recipe by multiplying the whole ratio, never by adding.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 开始 with ten. When it goes well, add more.

Revise a fraction of a whole — a ratio 2 : 3 splits the whole into fifths and that is the base. Write only the number.

Practise the questions

On paper: ratios around you

Always write ratios with a name of what the first and the second number are.

Simplify to simplest form: 15 : 25, 36 : 24, 100 : 75. Write what you divide by.
A pancake recipe has flour and milk in the ratio 2 : 5. How much milk belongs with 6 spoons of flour? Expand the ratio and write the method.
In the class there are 14 boys and 21 girls. Write the ratio of boys to girls in simplest form and say what fraction of the class the boys are.
Write as a ratio and simplify: 30 minutes to 2 hours. Watch the units.

Now you 💪

I read a ratio with the things and I know the order of the numbers must not be swapped.
I can simplify and expand a ratio and find simplest form.
Before writing a ratio I match the units.

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Done when: All ratios are in simplest form, the pancakes have 15 spoons of milk and the time ratio came out 1 : 4.

What to take from this lesson

A ratio = a recipe: how many parts of what. You choose the size of a part.
Order in a ratio is binding — read it with the things, not just the numbers.
You may simplify and expand only by multiplying/dividing both numbers.
A ratio a : b splits the whole into a + b parts; the first has a/(a+b) of the whole.
Match the units before you write the colon.