Answer :
To solve the problem of determining by what percentage the true weight of the goods is being marked up by the shopkeeper shifting the fulcrum, we need to follow these steps:
1. Understand the Setup:
The balance consists of two pans that are 39.3 cm apart, with the fulcrum initially in the center (which is 19.65 cm from each pan). However, the dishonest shopkeeper has shifted the fulcrum 1.46 cm to one side.
2. Determine Distances:
- Calculate the original distance from the fulcrum to each pan if the fulcrum were in the center:
[tex]\[
\text{Original Distance} = \frac{39.3}{2} = 19.65 \, \text{cm}
\][/tex]
- After shifting the fulcrum 1.46 cm, the new distances will be:
- Left Side Distance: The side closer to the fulcrum will be:
[tex]\[
19.65 - 1.46 = 18.19 \, \text{cm}
\][/tex]
- Right Side Distance: The side that is now farther from the fulcrum will be:
[tex]\[
19.65 + 1.46 = 21.11 \, \text{cm}
\][/tex]
3. Calculate Weight Comparison:
- The true weight of the goods corresponds to the left side (shorter distance).
- The marked or perceived weight by the customer is proportional to the right side (longer distance).
4. Calculate the Ratio of Marked to True Weight:
The ratio is determined by dividing the longer distance (right side) by the shorter distance (left side):
[tex]\[
\text{Weight Ratio} = \frac{21.11}{18.19} \approx 1.1605
\][/tex]
5. Determine the Percentage Markup:
- To find the percentage markup, subtract 1 from the weight ratio (to find the increase over the true weight), and multiply by 100:
[tex]\[
\text{Markup Percentage} = (1.1605 - 1) \times 100 = 16.05\%
\][/tex]
Thus, the shopkeeper is marking up the true weight of the goods by approximately 16.05%.
1. Understand the Setup:
The balance consists of two pans that are 39.3 cm apart, with the fulcrum initially in the center (which is 19.65 cm from each pan). However, the dishonest shopkeeper has shifted the fulcrum 1.46 cm to one side.
2. Determine Distances:
- Calculate the original distance from the fulcrum to each pan if the fulcrum were in the center:
[tex]\[
\text{Original Distance} = \frac{39.3}{2} = 19.65 \, \text{cm}
\][/tex]
- After shifting the fulcrum 1.46 cm, the new distances will be:
- Left Side Distance: The side closer to the fulcrum will be:
[tex]\[
19.65 - 1.46 = 18.19 \, \text{cm}
\][/tex]
- Right Side Distance: The side that is now farther from the fulcrum will be:
[tex]\[
19.65 + 1.46 = 21.11 \, \text{cm}
\][/tex]
3. Calculate Weight Comparison:
- The true weight of the goods corresponds to the left side (shorter distance).
- The marked or perceived weight by the customer is proportional to the right side (longer distance).
4. Calculate the Ratio of Marked to True Weight:
The ratio is determined by dividing the longer distance (right side) by the shorter distance (left side):
[tex]\[
\text{Weight Ratio} = \frac{21.11}{18.19} \approx 1.1605
\][/tex]
5. Determine the Percentage Markup:
- To find the percentage markup, subtract 1 from the weight ratio (to find the increase over the true weight), and multiply by 100:
[tex]\[
\text{Markup Percentage} = (1.1605 - 1) \times 100 = 16.05\%
\][/tex]
Thus, the shopkeeper is marking up the true weight of the goods by approximately 16.05%.