The Balance Table With a Tipped Pan
In the science corner, Ms. Okafor has set out three balance cards. Each card shows what happened when objects were placed on a pan balance.
Card A (level): 1 cube + 2 marbles on the left balances 8 marbles on the right. Card B (level): 1 cube + 1 cylinder on the left balances 10 marbles on the right. Card C (not level): 2 cylinders on the left against 6 marbles on the right, left pan lower.
All cubes are identical, all cylinders are identical, all marbles are identical. Students must answer without putting anything on a real balance. Daya writes "2 cylinders = 6 marbles" for Card C. Ms. Okafor marks it wrong and asks the class why that mistake is easy to make. The last question on the sheet is: how many marbles would balance 2 cubes and 3 cylinders? Every substitution must be shown.
1. Find the value of one cube and one cylinder in marbles. Explain which card you start with and why.
Read the worked answer
One cube = 6 marbles. One cylinder = 4 marbles.
Start with Card A, because it is the only level card holding a single unknown kind of object next to a known one.
- Card A: cube + 2 marbles = 8 marbles. Take the 2 marbles off both sides: cube = 8 − 2 = 6 marbles.
- Card B: cube + cylinder = 10 marbles. Put 6 in for the cube: 6 + cylinder = 10, so cylinder = 4 marbles.
Check against Card C: 2 cylinders would be 8 marbles against 6 marbles, and 8 > 6, so the left pan should indeed be lower. It is.
Order matters: Card B has two unknowns, so it cannot be solved until Card A produces a number to put into it.
2. Why is Daya's equation from Card C wrong, and what does Card C actually tell you?
Read the worked answer
A level balance gives an equation. A tipped balance does not.
An equals sign means both pans hold the same amount. On Card C the left pan is lower, which means the left side is heavier. Writing "2 cylinders = 6 marbles" throws away the very information the card was showing.
What Card C really says: 2 cylinders weigh more than 6 marbles, so one cylinder weighs more than 3 marbles.
That is still useful, but as a check, not a solution. Our cylinder is 4 marbles, and 4 > 3, so the card agrees with us. It could never have been used to find the 4, because "more than" allows many possible weights.
The easy mistake is treating every balance picture as an equals sign. The tilt is the clue.
3. How many marbles balance 2 cubes and 3 cylinders? Show each substitution and one way to check the answer.
Read the worked answer
24 marbles.
Substitute the values found earlier: cube = 6 marbles, cylinder = 4 marbles.
- 2 cubes = 2 × 6 = 12 marbles
- 3 cylinders = 3 × 4 = 12 marbles
- 12 + 12 = 24 marbles
Check by rebuilding Card B: 1 cube + 1 cylinder = 6 + 4 = 10 marbles, which matches the card. Check Card A: 1 cube + 2 marbles = 6 + 2 = 8 marbles, which matches. Check Card C: 2 cylinders = 8 marbles, which is more than the 6 on the other pan, so the left pan being lower is correct.
All three cards agree with the numbers, which is the strongest kind of check available here - each card tests the answer a different way.