Two Summer Savings Plans
Maya and her cousin Leo are saving for a summer trip. Maya starts with $60 and adds $15 each week from her part-time job. Leo starts with $100 and, because of a savings account promotion, his balance increases by 10% each week. Their balances after w weeks can be modeled as M(w) = 60 + 15w and L(w) = 100(1.1)^w. After four weeks, Maya says, 'I am adding more dollars per week than Leo, so I will catch up soon.' Leo's balance after 0, 1, 2, and 3 weeks is $100, $110, $121, and $133.10. The trip costs $300 per student, and both need to reach that amount before the deposit deadline in 12 weeks. Their teacher asks them to compare the plans using average rates of change, decide whether Maya's catch-up claim is supported, and determine who reaches $300 first.
1. Find each person's average rate of change in savings over weeks 0 to 4. Interpret the comparison.
Read the worked answer
Average rate of change = (change in balance)/(change in weeks).
For Maya over weeks 0 to 4: M(4) = 60 + 15(4) = 120, so (120 - 60)/4 = 15 dollars per week.
For Leo: L(4) = 100(1.1)^4 = 146.41, so (146.41 - 100)/4 = 11.60 dollars per week.
Maya's balance is growing faster in average dollars per week during this interval, but Leo still has the larger balance: $146.41 versus $120. The average rate alone does not determine who is ahead.
2. Is Maya's claim that she will catch up soon supported by the models? Use values through week 12.
Read the worked answer
Maya's claim is not supported. After 4 weeks she has $120 and Leo has $146.41. By week 8, Maya has 60 + 15(8) = $180; Leo has 100(1.1)^8 ≈ $214.36. By week 12, Maya has 60 + 15(12) = $240; Leo has 100(1.1)^12 ≈ $313.84.
Maya's lead in weekly dollar increase is not enough to close the $40 starting gap within 12 weeks. Leo's 10% growth applies to a growing balance, so his dollar increase per week eventually becomes larger than $15. Maya's reasoning looks only at the early weekly increase and ignores the starting difference and compounding.
3. Who reaches $300 first? Show the reasoning for each plan.
Read the worked answer
Maya reaches $300 when 60 + 15w = 300, so 15w = 240 and w = 16 weeks.
Leo's balance crosses $300 between weeks 11 and 12: L(11) = 100(1.1)^11 ≈ $285.31 and L(12) ≈ $313.84. So Leo reaches $300 first, during week 12, while Maya would need 16 weeks.
If the deposit is due at 12 weeks, Leo can make it and Maya cannot under these plans.