Sarah And Josh (To Long to fit here)

Question

Sarah and Josh each start saving today. Sarah saves 30 cents every day. Josh saves 1 cent on the first day, 2 cents on the second day, 3 cents on the third day, and so on, saving one more cent each day. At the end of how many days will they have saved the same amounts?

Answer 1

Josh saves with an arithmetic series (each day the same amount (1 cent) is added.

 

S_n = (n/2)(2a₁ + (n - 1)d), where a₁ = first amount, n = number of days, d = common difference

S_n = (n/2)[(2(1) + (n - 1)1)]

S_{n =} (n/2)[2 + n - 1]

S_n = (n/2[n + 1]

 

Sarah just saves the same amount, 30, for each of the "n" days. When does

 

30n = (n/2)[n + 1] ?             Multiply both sides by 2

60n = n(n + 1)                      Distribute

60n = n² + n                         Subtract n from both sides

59n = n²                               Divide both sides by n

59 = n

 

Check:

 

59(30) = (59/)(2 + (n - 1)1)

1770 = (59/2)(60)

1770 = 1770

 

They have the same amount on the 59th day.