Answer :
Step-by-step explanation:
Let's solve each part step by step:
(i) Capacities of ten different vessels which can be filled completely by mug-A with exact number of times
Mug-A has a capacity of 50 ml. The capacities of vessels that can be filled completely by mug-A are multiples of 50 ml. The first ten such capacities are:
50
×
1
=
50
50×1=50 ml
50
×
2
=
100
50×2=100 ml
50
×
3
=
150
50×3=150 ml
50
×
4
=
200
50×4=200 ml
50
×
5
=
250
50×5=250 ml
50
×
6
=
300
50×6=300 ml
50
×
7
=
350
50×7=350 ml
50
×
8
=
400
50×8=400 ml
50
×
9
=
450
50×9=450 ml
50
×
10
=
500
50×10=500 ml
(ii) Capacities of ten different vessels which can be filled completely by mug-B with exact number of times
Mug-B has a capacity of 100 ml. The capacities of vessels that can be filled completely by mug-B are multiples of 100 ml. The first ten such capacities are:
100
×
1
=
100
100×1=100 ml
100
×
2
=
200
100×2=200 ml
100
×
3
=
300
100×3=300 ml
100
×
4
=
400
100×4=400 ml
100
×
5
=
500
100×5=500 ml
100
×
6
=
600
100×6=600 ml
100
×
7
=
700
100×7=700 ml
100
×
8
=
800
100×8=800 ml
100
×
9
=
900
100×9=900 ml
100
×
10
=
1000
100×10=1000 ml
(iii) Common capacities of vessels which can be filled completely by mugs A as well as B with exact number of times
The common capacities will be the least common multiples (LCM) of 50 ml and 100 ml, and multiples thereof. Since 100 is a multiple of 50, the LCM is 100 ml. Thus, the common capacities are multiples of 100 ml. The first ten such capacities are:
100 ml
200 ml
300 ml
400 ml
500 ml
600 ml
700 ml
800 ml
900 ml
1000 ml
(iv) The least capacity of a vessel which can be filled completely by mugs A as well as B with exact number of times
As discussed in part (iii), the least capacity is the LCM of 50 ml and 100 ml, which is:
LCM
(
50
ml
,
100
ml
)
=
100
ml
LCM(50 ml,100 ml)=100 ml
So, the least capacity of a vessel that can be filled completely by both mugs A and B is 100 ml.
