Week 02 — elder son, Grade 5 / Number
Making the pieces the same size — and cancelling back down
Last week was multiples and factors. Those were the tools. This week is what they are for.
The lowest common multiple is used to add fractions. The greatest common divisor is used to cancel them down. That is the whole story.
What is 1⁄4 ＋ 1⁄6 ?
1⁄4 ＋ 1⁄6 ＝ 2⁄10
Add the tops, add the bottoms. This is wrong.
Here is how to catch it without calculating. 1⁄4 is bigger than 1⁄6. We added those two together — and yet the answer 2⁄10 ＝ 1⁄5 is smaller than 1⁄4 was on its own. Adding something cannot make the total shrink.
One piece of a tape cut into 4, and one piece of a tape cut into 6, are pieces of different sizes.
Things of different sizes cannot simply be counted together. It is the same reason you cannot add 3 oranges and 2 apples and call the answer 5.
Quarters and sixths both become the same size if you cut everything into twelfths.
1⁄4 ＝ 3⁄12 1⁄6 ＝ 2⁄12
12 is the lowest common multiple of 4 and 6 — the smallest number that both denominators divide into.
This is the answer to 'why did we do lowest common multiples last week?'
18⁄24 can be written with fewer, bigger pieces.
The greatest common divisor of 18 and 24 is 6. Divide top and bottom by 6 and you get 3⁄4.
The size has not changed at all. Only the way it is written is shorter.
If finding a common denominator is cutting finer, cancelling is gathering back up. Same move, opposite direction.
1⁄4 ＋ 1⁄6
① LCM of 4 and 6 → multiples of 4: 4, 8, 12 / of 6: 6, 12 → 12
② 1⁄4: multiply top and bottom by 3 → 3⁄12 1⁄6: by 2 → 2⁄12
③ 3⁄12 ＋ 2⁄12 ＝ 5⁄12
Only the tops are added. The bottoms are not. Three twelfths and two twelfths make five twelfths.
5⁄6 − 1⁄4
① LCM of 6 and 4 is 12
② 5⁄6 ＝ 10⁄12 1⁄4 ＝ 3⁄12
③ 10⁄12 − 3⁄12 ＝ 7⁄12
Simplify 18⁄24
① Factors of 18: 1, 2, 3, 6, 9, 18 / of 24: 1, 2, 3, 4, 6, 8, 12, 24
② Largest shared factor: 6
③ 18 ÷ 6 ＝ 3 24 ÷ 6 ＝ 4 → 3⁄4
An answer is written in its simplest form, where nothing divides both parts.
1⁄6 ＋ 1⁄3
① LCM is 6 → 1⁄6 ＋ 2⁄6 ＝ 3⁄6
② 3 and 6 both divide by 3 → 1⁄2
After adding, always check whether the answer cancels. This is the step most often forgotten.
LCM of 2 and 3 is 6.
1⁄2 ＝ 3⁄6 1⁄3 ＝ 2⁄6
3⁄6 ＋ 2⁄6 ＝ 5⁄6
Not 2⁄5. Never add the denominators.
LCM of 4 and 6 is 12.
9⁄12 ＋ 2⁄12 ＝ 11⁄12
LCM of 6 and 9 is 18.
15⁄18 − 4⁄18 ＝ 11⁄18
GCD of 24 and 36 is 12.
24 ÷ 12 ＝ 2 36 ÷ 12 ＝ 3 → 2⁄3
Dividing by 2, 2, then 3 gets to the same place (12⁄18 → 6⁄9 → 2⁄3). Longer, but not wrong.
LCM of 4 and 12 is 12 — 12 is already a multiple of 4.
3⁄12 ＋ 1⁄12 ＝ 4⁄12
Both divide by 4 → 1⁄3
Common denominators are for comparing too, not only for adding.
LCM of 3 and 8 is 24.
2⁄3 ＝ 16⁄24 5⁄8 ＝ 15⁄24
16 > 15, so 2⁄3 is larger.
This asks for a sentence, not a number.
A good answer:
"1⁄2 on its own is already bigger than 2⁄5. Adding 1⁄3 to it cannot give an answer smaller than what we started with. Adding the denominators changes how big one piece is, which is why it is not allowed."
Being able to say this is the most important thing in the topic.
Both give a correct answer. Using 48 makes the numbers larger and leaves more cancelling to do at the end.
We use the lowest common multiple because it is the one least likely to need cancelling afterwards — not because it is a rule.
Now the maths takes off its maths clothes. Nothing underneath changes.
A computer screen is 1920 dots across and 1080 down. We call that screen 16 : 9.
The greatest common divisor of 1920 and 1080 is 120.
1920 ÷ 120 ＝ 16 1080 ÷ 120 ＝ 9
The shape of the screen has not changed at all. Only the writing got shorter. It is the identical move as turning 18⁄24 into 3⁄4.
A 12-tooth gear meshes with an 18-tooth gear. Both marks start pointing up.
After how many teeth do both marks point up again?
LCM of 12 and 18 → 36 teeth
By then the small gear has turned 36 ÷ 12 ＝ 3 times and the large one 36 ÷ 18 ＝ 2 times.
Not 12 × 18 ＝ 216. Multiplying counts how long until each has turned a whole number of times *the other's* count — far too far.
A recipe for four says 1⁄4 cup of sugar and 1⁄6 cup of salt. How much goes into the bowl in total?
1⁄4 ＋ 1⁄6 ＝ 5⁄12 of a cup.
No measuring cup has a 5⁄12 mark, so in practice you read it as "a little over 1⁄3". Fractions are for estimating an amount, not only for producing an exact answer.
These are not the real exam questions. The reason is in the box below each one, with a link to the official paper. Solve ours first, then open theirs alongside.
What this question is really testing：Not whether you can calculate, but whether you can say why the method works. Japan's national assessment asks this every year.
⑴ A good reason
"1⁄3 on its own is already bigger than 2⁄7. Adding 1⁄4 to it cannot produce 2⁄7."
The point is noticing by size, before any calculation.
⑵ LCM of 3 and 4 is 12.
4⁄12 ＋ 3⁄12 ＝ 7⁄12
⑶ A good explanation
"The mistake was adding the denominators. The denominator says how big one piece is, so adding them changes the size of the pieces. Make both denominators 12 first, then add only the numerators."
What this question is really testing：Indonesia's numerasi style. A fraction inside a real situation, with a spare number mixed in. Choosing which numbers to use is the task.
⑴ 1⁄4 ＋ 1⁄3. LCM of 4 and 3 is 12.
3⁄12 ＋ 4⁄12 ＝ 7⁄12
⑵ Full is 1 ＝ 12⁄12.
12⁄12 − 7⁄12 ＝ 5⁄12
⑶ No.
Both parts are answered in fractions. The 60 cm is never used.
Assuming every number given must be used is exactly the trap. AKM and PISA put spare numbers in on purpose.
(Had the question asked for centimetres: 7⁄12 × 60 ＝ 35 cm, with 5⁄12 × 60 ＝ 25 cm still to go. Then it would have been needed.)