Find two numbers such that their mean proportion is 16 and third proportion is 1024.
- A4 and 32
- B4 and 64
- C8 and 64
- D8 and 32
Solution & Step-by-step Explanation
Let the two numbers be a and b.
Mean Proportion Condition:
The mean proportion of a and b is given by
ab
=16.
Squaring both sides:
ab=16
2
=256⟹a=
b
256
Third Proportion Condition:
The third proportion of a and b is given by
a
b
2
=1024.
Substitute a=
b
256
into this formula:
b
256
b
2
=1024
256
b
3
=1024
b
3
=1024×256
Expressing as powers of 2:
1024=2
10
256=2
8
b
3
=2
10
×2
8
=2
18
b=(2
18
)
1/3
=2
6
=64
Finding a:
a=
b
256
=
64
256
=4
Thus, the two numbers are 4 and 64.
Mean Proportion Condition:
The mean proportion of a and b is given by
ab
=16.
Squaring both sides:
ab=16
2
=256⟹a=
b
256
Third Proportion Condition:
The third proportion of a and b is given by
a
b
2
=1024.
Substitute a=
b
256
into this formula:
b
256
b
2
=1024
256
b
3
=1024
b
3
=1024×256
Expressing as powers of 2:
1024=2
10
256=2
8
b
3
=2
10
×2
8
=2
18
b=(2
18
)
1/3
=2
6
=64
Finding a:
a=
b
256
=
64
256
=4
Thus, the two numbers are 4 and 64.