The Shopkeeper's Unequal-Arm Balance
During the annual science-club audit of the local market, Class 9 students bring a shopkeeper's beam balance to the school laboratory. They have no certified 1 kg standard, but the shopkeeper's brass weights were checked against the school's standard set the previous month. A bag of rice placed in pan A balances exactly against 1.44 kg in pan B. The same bag placed in pan B balances against 1.00 kg in pan A. In normal business the shopkeeper always puts the goods in pan A and the weights in pan B. Each weighing is repeated three times with identical results. With the pans empty the beam is not level unless a small loop of wire is hung on pan B; the students remove this wire and re-level the pointer before every reading. The shopkeeper argues that the two weighings average to 1.22 kg, so the bag weighs more than 1 kg and the customer is actually favoured. The club must settle the question with evidence and arithmetic.
1. Using only these two weighings, find the true mass of the rice and the ratio of the two arm lengths. Then decide what a customer who asks for 1.00 kg of rice actually receives.
Read the worked answer
Let the arm lengths be L(A) and L(B) and the true mass be m. A beam balance gives equal torques:
- Rice in A: m x L(A) = 1.44 x L(B)
- Rice in B: m x L(B) = 1.00 x L(A)
Multiply the two equations: m^2 x L(A) x L(B) = 1.44 x L(A) x L(B), so m^2 = 1.44 and m = 1.20 kg.
Divide the first by the second: L(A)/L(B) = 1.44/1.00 x L(B)/L(A), giving (L(A)/L(B))^2 = 1.44, so L(A)/L(B) = 1.20. Pan A's arm is 20% longer.
Now the sale. Goods go in A, the 1.00 kg standard in B: m' x L(A) = 1.00 x L(B), so m' = L(B)/L(A) = 1/1.20 = 0.833 kg. The customer asking for 1 kg receives about 0.83 kg and is short by 0.167 kg, that is one-sixth or 16.7% of what was paid for. The longer arm is on the goods side, which is exactly the arrangement that cheats the buyer.
2. The shopkeeper says the two weighings average to 1.22 kg, so the customer gains. Explain with the balance condition why the arithmetic mean is wrong here, and why the geometric mean is the correct combination.
Read the worked answer
The two readings are not equal errors in opposite directions; they are reciprocals of each other. If r = L(A)/L(B), then the reading with rice in A is m/r and the reading with rice in B is m x r. Their product is m^2 regardless of r, which is why the geometric mean is right: sqrt(1.44 x 1.00) = 1.20 kg.
The arithmetic mean is m(r + 1/r)/2. Since r + 1/r is always at least 2 (equality only when r = 1), the mean is always greater than or equal to m; the error never cancels. With r = 1.20: (1.20 + 0.8333)/2 = 1.0167, and 1.20 x 1.0167 = 1.22 kg. So the mean overshoots the truth by 0.02 kg, about 1.7%.
The shopkeeper's argument fails twice over. The mean is biased upward, and even if the true mass were 1.22 kg, the customer is still handed only 1.20/1.20 = 1.00 kg? No: the goods actually delivered weigh 0.833 kg per kilogram charged, whatever the bag's true mass.
3. Describe two quick checks the students could perform in the market, with no certified mass available, that would expose a balance with unequal arms.
Read the worked answer
Weigh the same object twice, once in each pan. With equal arms the two readings must be identical. If they read 1.44 kg and 1.00 kg, the ratio of readings is (L(A)/L(B))^2, so the arms differ, and the object's true mass is the geometric mean of the readings. This needs no standard at all, only the shopkeeper's own weights.
Swap the pans themselves (or turn the beam end for end) with the balance empty and see whether the pointer still rests at zero. The students had already found a wire loop on pan B, which is a zero error that must be corrected before any comparison; after removing it and re-levelling, a persistent tilt on swapping would show the beam itself is asymmetric.
A third check: weigh a known volume of water, since 1 litre of water has a mass of about 1 kg. A measuring bottle filled to 1 L and weighed should give 1.00 kg on a true balance.
Any one of these settles the matter without a certified standard.