Number
Properties of rational numbers
Add, subtract, multiply and divide rational numbers, and use the properties that make the work shorter.
You have met rational numbers already: anything you can write as one whole number over another. What comes next is the set of rules they obey — the properties. They are worth knowing not because they are impressive names, but because each one lets you do less work.
The properties, in plain words
Order does not matter when you add or multiply (commutative). Grouping does not matter either (associative). Subtraction and division break both of those rules. Adding zero changes nothing, and multiplying by one changes nothing. Every rational number has an opposite that adds to zero, and every one except zero has a reciprocal that multiplies to one.
Have a play
Wander along this line in quarter steps. Nothing here is right or wrong — see how the numbers crowd together.
Tap anywhere on the line.
Worked example
Work out 7/5 × 3/4 + 7/5 × 1/4 without doing two multiplications.
Notice that 7/5 multiplies both parts.
The distributive property says a × b + a × c is the same as a × (b + c). Spotting the shared factor is the whole trick.
Try it together
Now let us do -3/7 × 5/4 + -3/7 × 3/4 together.
Same shape as the example, with a negative common factor. Take the shared factor out, add what is left, then multiply once.
1.Both products are multiplied by the same fraction. Write it.
Have a go
Have a go on your own: 5/8 × 4/3 + 5/8 × 2/3 = ? Write your answer as a fraction.
Hint: Take out the shared factor first. The two fractions left inside add to a whole number.
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.