Q96. With the use of three different weights of 1kg, 3kg and 9kg, how many objects of different weights can be weighed, if the objects to be weighed and the given weights may be placed in either pan of scale?
(a) 15
(b) 13
(c) 11
(d) 9
Answers
ANSWER:
Given:
- 3 weights: 1kg, 3kg and 9kg.
To Find:
- Number of objects of different weights can be weighed, if the objects to be weighed and the given weights may be placed in either pan of scale.
Solution:
We are given three weights of 1kg, 3kg and 9kg.
To find the number of objects that can be weighed, we take various situations. And after that, to obtain the total amount, we will add all the weighings that we got in each situation.
1st Situation: Taking All Weights Individually.
⇒ There can be 3cases in this situation.
1. Taking 1kg weight,
2. Taking 3kg weight, and
3. Taking 9kg weight.
Hence, there will be 3 weighings possible in this situation.
Therefore,
⇒ No of weighings = 3
2nd Situation: Taking 2 Weights at a time.
(we will keep the weights in the same pan of the scale.)
⇒ There can be 3cases in this situation.
1. Taking 1kg and 3kg weights,
2. Taking 3kg and 9kg weights, and
3. Taking 9kg and 1kg weights.
Hence, there will be 3 weighings possible in this situation.
Therefore,
⇒ No of weighings = 3
3rd Situation: Taking 2 Weights at a time.
(we will keep the weights in different pans of the scale.)
⇒ There can be 3cases in this situation.
1. Taking 1kg and 3kg weights,
2. Taking 3kg and 9kg weights, and
3. Taking 9kg and 1kg weights.
Hence, there will be 3 weighings possible in this situation.
Therefore,
⇒ No of weighings = 3
4th Situation: Taking 3 Weights at a time.
(we will keep the weights in the same pan of the scale.)
⇒ There can be 1 case in this situation.
1. Taking 1kg, 3kg and 9kg weights.
Hence, there will be 1 weighing possible in this situation.
Therefore,
⇒ No of weighings = 1
5th Situation: Taking 3 Weights at a time.
(we will keep the weights in different pans(2 in one pan, the 3rd in the other) of the scale.)
⇒ There can be 3cases in this situation.
1. Taking 1kg and 3kg weights in 1 pan and 9kg weight in other,
2. Taking 3kg and 9kg weights in 1 pan and 1kg weight in other, and
3. Taking 9kg and 1kg weights in 1 pan and 3kg weight in other.
Hence, there will be 3 weighings possible in this situation.
Therefore,
⇒ No of weighings = 3
Now, as mentioned above, we will add all the no. of possible weighings.
So,
⇒ Total weighings = (Weighing in 1st situation) + (Weighing in 2nd situation) + (Weighing in 3rd situation) + (Weighing in 4th situation) + (Weighing in 5th situation)
So,
⇒ Total weighings = 3 + 3 + 3 + 1 + 3
Hence,
⇒ Total weighings = 13
Therefore, 13 objects of different weights can be weighed if the objects to be weighed and the given weights may be placed in either pan of scale.