Self-Help Group Savings Target
A women's self-help group in a village near Berhampur plans to buy a sewing machine costing ₹10,000. The group starts a monthly savings scheme: it deposits ₹200 in the first month and increases the deposit by ₹50 every month. The treasurer claims that the machine can be bought after 15 months. The group also has an offer from a bank: if the scheme runs for 20 months, the bank will add a bonus of ₹500. The members want to know the total savings after 15 months, the first month in which savings alone reach ₹10,000, and whether waiting for 20 months with the bonus gives at least ₹1,000 more than buying at the earliest possible month. The group cannot skip deposits and cannot withdraw before the target month.
1. Find the total savings after 15 months and the shortfall against the ₹10,000 target.
Read the worked answer
Deposits form an AP with a=200, d=50. S_15 = 15/2 [2×200 + 14×50] = 7.5(400+700) = 7.5×1100 = ₹8,250. Shortfall = 10,000 − 8,250 = ₹1,750. So the treasurer's 15-month claim is wrong.
2. Find the first month in which the savings alone reach ₹10,000.
Read the worked answer
S_n = n/2[400+(n−1)50] = n/2[350+50n] = 25n² + 175n. Set 25n² + 175n ≥ 10,000, i.e. n² + 7n − 400 ≥ 0. At n=16: S=25(256)+175(16)=6,400+2,800=₹9,200. At n=17: S=25(289)+175(17)=7,225+2,975=₹10,200. First month is the 17th.
3. Compare the 20-month plan with the ₹500 bonus against buying at the earliest possible month. Is the difference at least ₹1,000?
Read the worked answer
At 20 months: S_20 = 20/2[400+19×50] = 10(400+950) = ₹13,500. With ₹500 bonus: ₹14,000. Earliest target month is 17 with ₹10,200. Difference = 14,000 − 10,200 = ₹3,800, which is more than ₹1,000. Waiting is financially better if the machine can be delayed.