Problem:
The least number of complete years in which a sum of money put out at 10% compound interest will be more than doubled is:

a) 8 years
b) 6 years
c) 4 years
d) 7 years

Solution:

To solve this problem, we need to find the number of years it takes for a sum of money to double at a compound interest rate of 10%.

Formula:
The formula to calculate the compound interest is:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested or the loan is taken for

In this case, we need to find the value of t.

Given:
r = 10% = 0.10
P = 1 (as we want to double the money)
A = 2 (as we want the money to be more than doubled)

We can rewrite the formula as:
2 = 1(1 + 0.10/n)^(n*t)

Now, we need to check each option to find the least number of years it takes for the money to double.

Option a) 8 years:
In this case, t = 8
2 = 1(1 + 0.10/n)^(8*n)

Option b) 6 years:
In this case, t = 6
2 = 1(1 + 0.10/n)^(6*n)

Option c) 4 years:
In this case, t = 4
2 = 1(1 + 0.10/n)^(4*n)

Option d) 7 years:
In this case, t = 7
2 = 1(1 + 0.10/n)^(7*n)

Calculations:
To compare the options, we can simplify the equation by assuming different values for n and evaluating the right-hand side of the equation.

For example, let's assume n = 1 (compounded annually):
2 = 1(1 + 0.10/1)^(8*1)
2 = 1(1 + 0.10)^8
2 = 1(1.10)^8
2 = 1(1.10^8)
2 = 1(2.1436)
2 = 2.1436 (approximately)

This shows that with n = 1, the money does not double in 8 years.

By comparing the options using different values of n, we find that the money doubles in the least number of years with option a) 8 years. Therefore, the correct answer is option a) 8 years.