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How would you go about solving this?

Robert and Daniel both pay money into a joint savings account each month. Robert pays in £r each month and Daniel pays in £d each month. They both decide to increase their monthly contributions by 6%. Robert says that this will increase their joint contribution by 12%, but Daniel says it will only increase it by 6%.
Who is correct: Robert, Daniel or neither of them? Justify your answer using algebra.

Robert pays in £r and Daniel pays in £d into a joint savings account each month. Therefore, the total paid into the account (in pounds) each month can be expressed as:

[tex]\textsf{Joint monthly contribution}=r + d[/tex]

They both decide to increase their monthly contributions by 6%.

When a number is increased by 6%, we add 6% to the original amount. As the original amount represents 100%, the new total becomes 106%. Therefore, to increase each of the original payments by 6%, we multiply each of them by 1.06:

[tex]\textsf{Robert's new monthly payment}=r \times 1.06 = 1.06r[/tex]

[tex]\textsf{Daniels new monthly payment}=d \times 1.06 = 1.06d[/tex]

Therefore, the new total paid into the account (in pounds) each month is the sum of the new individual monthly payments:

[tex]\textsf{New joint monthly contribution}=1.06r + 1.06d[/tex]

Factor out 1.06:

[tex]\textsf{New joint monthly contribution}=1.06(r + d)[/tex]

Therefore, we can see that the original joint monthly contribution (r + d) has been increased by 6%, and so Daniel is correct.