A fraction is a division
Every fraction is a division you haven't done yet. means “3 divided by 4”, which is 0.75. The bottom number (the denominatorThe bottom of a fraction: how many equal parts the whole is cut into.) says how many equal pieces the whole is cut into; the top (the numeratorThe top of a fraction: how many of the pieces you take.) says how many you take.
The same amount can wear many fractions. Cut a pizza into 4 slices and take 3, or into 8 slices and take 6: you ate the same pizza. Multiplying the top and bottom by the same number never changes a fraction's value.
Ratios: recipes that scale
A ratioA comparison of two amounts, written a : b. It describes a recipe, not a total. compares two amounts. A paint recipe of “2 parts blue to 3 parts yellow” is written . It says nothing about how much paint you make, only about how it's mixed. Try it:
Tins of blue and yellow paint are poured into a bucket. Sliders set how many parts of blue and of yellow go into one batch, and how many batches you make. The bucket's colour depends only on the fraction of blue, so making more batches of the same recipe keeps the colour; changing the recipe changes it.
Mix the paint
InteractiveThe math, with your numbers
More batches multiply the top and the bottom by the same number, so the fraction (and the colour) stays put.
Try this
- →Set 2 blue : 3 yellow, then add batches. Does the colour change?
- →Find two different recipes that give exactly 50% blue.
- →Which is bluer: 3 : 4 or 4 : 6? Predict first, then check the percentages.
Making three batches means tins, yet the colour is identical. That's the fraction rule again: out of every 5 tins, 2 are blue, so blue is always of the mix, whether you pour 5 tins or 500.
Worked example: 20 litres of the 2 : 3 green
Step 1 of 4- 1Add up the parts: 2 + 3 = 5 parts in every batch.
Percentages are fractions out of 100
“Per cent” is Latin for “out of a hundred”. A percentage is just a fraction with 100 on the bottom, which makes different fractions easy to compare.
Say it out loud: “divide the part by the whole, then multiply by a hundred.”
Part
The piece you're interested in (the blue tins).
Whole
Everything there is (all the tins).
Make a prediction first
Which mix is bluer: 3 : 4 or 4 : 6?
Think about it, then click to reveal the answer.
Make a prediction first
Which mix is bluer: 3 : 4 or 4 : 6?
Think about it, then click to reveal the answer.
Turn each into a fraction of the total. 3 : 4 has 7 parts, so blue is . 4 : 6 has 10 parts, so blue is . The 3 : 4 mix is (just) bluer, even though it uses fewer blue tins. Check it in the simulation.
'Per' means divide
The most useful word in science is “per”. Every time you see it, there's a division hiding:
A quantity “per” something is called a rateAn amount of one thing for each single unit of another, like metres per second. Found by dividing.. Rates let you compare things of different sizes fairly: a car that goes 150 km in 2 hours (75 km per hour) is faster than one that goes 200 km in 3 hours (about 67 km per hour), even though the second went further.
Worked example: which bottle of juice is the better deal?
Step 1 of 3- 1Bottle A: 1.5 litres for $3. Bottle B: 2 litres for $3.60. Find the price per litre of each.
Wrap-up
Key ideas to remember
- A fraction is a division. Multiplying top and bottom by the same number doesn't change it.
- A ratio is a recipe. To get each share, divide the total by the sum of the parts.
- A percentage is a fraction out of 100: part ÷ whole × 100%.
- “Per” means “divided by”. Rates like m/s or kg/m³ compare things of different sizes fairly.
Check your understanding
Sources & further reading
Everything in this lesson agrees with these references. They're all free to read.