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Balance scales

Algebra · level 1, lesson 2 of 5

A question to start

Look at this scale. One apple on one side, three blocks on the other side, and the scale is level.

1 apple balances 3 blocks

Now put a second apple, just like the first one, on the left side. How many blocks do you need on the right side to make the scale level again?

Think about it before you read on. The answer is not 4. (It is in the worked examples.)

The idea

What a balance scale does

A balance scale is a bar that rests on a point in the middle, with a pan hanging from each end. You put things on the pans. The scale does not tell you a number. It only answers one question: which side is heavier?

  • If both sides weigh the same, the bar stays flat. We say the scale is level, or balanced.
  • If one side is heavier, that side goes down, and the lighter side goes up.

That is all a balance does. It compares. It is like two children on a see-saw: the heavier child sinks, the lighter child rises, and two children of the same weight can sit still in the air.

A heavier side goes down

Here are 3 blocks on the left and 5 blocks of the same kind on the right. All the blocks weigh the same. The right side has more blocks, so it is heavier, and it goes down.

5 blocks are heavier than 3 blocks: the right side goes down

Watch out: the lower side is the heavier side. Some children think the side that goes up "wins". On a balance, going down means heavier.

Adding to one side tips it

Start with a level scale: 3 blocks on each side. Now add 1 more block to the left side only. The left side now has 4 blocks and the right side has 3. The left side is heavier, so it goes down.

Why? Before, the two sides were the same. Then the left side got something extra and the right side got nothing. So the left side now has more than the right. Adding to just one side always breaks the balance.

Taking something off one side also breaks the balance. If you take a block off the right side, the right side gets lighter and goes up.

Adding the same to both sides keeps it level

Start again with a level scale: 3 blocks on each side. Now add 2 blocks to the left and 2 blocks to the right.

3 and 2 more on each side: still level

The scale stays level. Why? The two sides were the same weight before. Then both sides got exactly the same extra weight. "The same, plus the same extra" is still the same. Nothing happened to make one side heavier than the other.

Here is a way to think about it. Imagine two friends with the same number of stickers. Their grandma gives each of them 2 more stickers. Do they still have the same number? Yes! Each of them got the same present.

The same is true for taking away. If a level scale has 5 blocks on each side, and you take 1 block off each side, it still balances: 4 and 4.

This is the golden rule of the balance:

If you do the same thing to both sides, a level scale stays level.

This rule is the most important rule in all of algebra. You will use it in every level, all the way to the top.

Things that weigh the same can swap places

Go back to the apple. One apple balances 3 blocks. That tells us: one apple weighs the same as 3 blocks. So anywhere you see an apple, you could take it off and put 3 blocks there instead, and nothing changes. The scale does not care what the things are, only how heavy they are.

That is how we can solve the starting puzzle. Two apples are like two lots of 3 blocks. That is 3 blocks and 3 more blocks: 6 blocks.

Finding a mystery weight

Now a bag. We do not know what is inside. But we know the bag holds some blocks, all the same kind as the loose blocks. The scale shows a bag and 2 blocks on the left, and 5 blocks on the right. It is level.

? a bag and 2 blocks balance 5 blocks

How many blocks are in the bag? Use the golden rule. Take 2 blocks off the left. To keep the scale level, take 2 blocks off the right too. Now the bag is alone on the left, and 3 blocks are on the right. The scale is still level.

? take 2 off each side: the bag balances 3 blocks

So the bag holds 3 blocks. Check: 3 blocks in the bag and 2 loose ones make 5 blocks, the same as the right side. ✓

This is exactly what algebra does. The bag is the mystery number. The scale is the equal sign. And "do the same to both sides" is how you find the mystery number.

Making a tipped scale level

If a scale is tipped, you can make it level by adding to the lighter side (the side that is up) or taking from the heavier side (the side that is down). With 3 blocks on the left and 5 on the right, you could:

  • add 2 blocks to the left (3 and 2 more is 5, the same as the right), or
  • take 2 blocks off the right (5 take away 2 is 3, the same as the left).

Both ways work. The difference between the sides is 2 blocks, so you need to fix 2 blocks.

A balance on Mars

On Mars everything weighs much less than on Earth: a heavy backpack feels light there. Would a balance scale still work? Yes! The things on the left pan get lighter, and the things on the right pan get lighter in exactly the same way. If they were the same on Earth, they are still the same on Mars. A balance compares the two sides, so it gives the same answer on any planet. (A bathroom scale is different: it would show a much smaller number on Mars.)

Worked examples

1. The apples. One apple balances 3 blocks. How many blocks balance 2 apples?

Each apple weighs the same as 3 blocks. So 2 apples weigh the same as 3 blocks and 3 more blocks. Count them: 3 and 3 make 6 blocks. Check: put each apple next to its own 3 blocks. Every apple has a matching pile, so the sides are the same.

Why not 4? Adding 1 apple to the left adds the weight of 3 blocks, not 1 block. To keep it level, the right side needs the same weight added, which is 3 blocks: 3 and 3 make 6.

2. Which way does it tip? Left: 4 blocks. Right: 6 blocks. All blocks are the same.

The right side has more blocks, so it is heavier and goes down. The left side goes up. To make it level, add 2 blocks to the left, or take 2 off the right.

3. Same to both sides. A scale is level with 4 blocks on each side. Mia adds 3 blocks to each side. Then she takes 1 block off each side. Is it still level?

Yes. Each time she did the same thing to both sides. Left: 4, then 7, then 6. Right: 4, then 7, then 6. Level the whole time.

4. A mystery bag. A bag and 1 block balance 4 blocks. How many blocks are in the bag?

Take 1 block off each side. The bag balances 3 blocks, so the bag holds 3. Check: 3 and 1 make 4 ✓.

5. Two bags. Two bags, exactly the same, balance 6 blocks. How many blocks are in each bag?

The two bags share the weight of 6 blocks equally. Put the 6 blocks into two equal piles: 3 and 3. Each bag balances one pile, so each bag holds 3 blocks. Check: 3 and 3 make 6 ✓.

Try it

  1. A scale has 2 blocks on the left and 7 blocks on the right. Which side goes down?
  2. A scale is level with 5 blocks on each side. You add 1 block to the right only. What happens?
  3. A scale is level with 3 blocks on each side. You add 4 blocks to each side. Is it still level? How many blocks are on each side now?
  4. A scale has 6 blocks on the left and 4 on the right. Give two different ways to make it level.
  5. A bag and 2 blocks balance 6 blocks. How many blocks are in the bag?
  6. One pear balances 2 blocks. How many blocks balance 3 pears?
  7. On the left: a bag and 3 blocks. On the right: a bag just like it (with the same number inside) and 5 blocks. Can this scale ever be level? (Hint: take a bag off each side. What is left?)
  8. Two equal bags and 1 block balance 9 blocks. How many blocks are in each bag?
Answers
  1. The right side goes down, because 7 blocks are heavier than 2 blocks.
  2. The right side gets heavier, so the right goes down and the left goes up. Adding to only one side breaks the balance.
  3. Yes, it is still level, because you did the same to both sides. Each side now has 3 and 4 more, which is 7 blocks.
  4. The left is heavier by 2 blocks. You can add 2 blocks to the right (4 and 2 more is 6), or take 2 blocks off the left (6 take away 2 is 4). Another way: take 1 off the left and add 1 to the right, so both have 5.
  5. 4 blocks. Take 2 blocks off each side: the bag balances 4 blocks. Check: 4 and 2 make 6 ✓.
  6. 6 blocks. Each pear is worth 2 blocks: 2, 4, 6. Check: put 2 blocks next to each pear; three pears, three pairs of blocks, 6 blocks.
  7. It is not possible. Take the same bag off each side (the golden rule keeps it level). Then 3 blocks would have to balance 5 blocks, and they do not: 5 is heavier. So a scale like that could never be level. Spotting that a puzzle has no answer is good detective work too.
  8. 4 blocks in each bag. Take 1 block off each side: two bags balance 8 blocks. Share 8 into two equal piles: 4 and 4. Check: 4 and 4 and 1 make 9 ✓.

Common mistake

Thinking the side that goes up is heavier. On a see-saw, the child who rises into the air is the lighter one. On a balance, the side that sinks is the heavy side.

Doing something to only one side. To find what is in a bag, some people take 2 blocks off the side with the bag and forget the other side. Then the scale tips, and the answer is wrong. Whatever you do, do it to both sides.

Where it leads