Question

1-72 ramon is saving $7.75 per week to buy a new cell phone. the phone he wants costs $125.00. for how many weeks will he need to save his money? first, estimate your answer. then figure out the actual number of weeks.

Explanation:

Step1: Estimate the number of weeks

To estimate, we can round \$7.75 to \$8. Then we divide the total cost (\$125) by the estimated weekly savings (\$8). So the estimation is \( \frac{125}{8} \approx 15.625 \), so we can estimate around 16 weeks (or 15 - 16 weeks as a rough estimate).

Step2: Calculate the actual number of weeks

We use the formula \( \text{Number of weeks} = \frac{\text{Total cost}}{\text{Weekly savings}} \). The total cost is \$125.00 and the weekly savings is \$7.75. So we calculate \( \frac{125}{7.75} \). Let's do the division: \( 125 \div 7.75=\frac{125}{\frac{31}{4}} = 125\times\frac{4}{31}=\frac{500}{31}\approx16.13 \). Since we can't have a fraction of a week in terms of saving (he needs to save at least enough), we need to round up to the next whole number because after 16 weeks, let's check: \( 16\times7.75 = 124 \), which is less than 125. After 17 weeks: \( 17\times7.75=131.75 \), which is more than 125. Wait, but actually, when calculating the number of weeks needed to reach at least \$125, we use the ceiling function. But let's do the exact division: LXI5. So we need to round up to 17? Wait, no, wait. Wait, the problem says "for how many weeks will he need to save his money". So if he saves \$7.75 per week, after 16 weeks he has \( 16\times7.75 = 124 \), which is less than 125. So he needs one more week, so 17? Wait, but let's check the division again. Wait, \( 7.75\times16 = 124 \), \( 125 - 124 = 1 \), so he needs an additional week to save that \$1. So the actual number of weeks is 17? Wait, no, wait, maybe I made a mistake. Wait, LXI9, so since he can't save a fraction of a week, he needs to save for 17 weeks? Wait, but let's check the estimation first. When we estimated, we used \$8 per week, \( 125\div8 = 15.625 \), so around 16 weeks, but the actual is a bit more. Wait, maybe the problem allows for decimal weeks? No, weeks are whole numbers. So we need to find the smallest integer \( n \) such that \( 7.75n\geq125 \). Solving for \( n \), \( n\geq\frac{125}{7.75}\approx16.13 \), so \( n = 17 \). Wait, but let's check: \( 7.75\times16 = 124 \), \( 7.75\times17 = 131.75 \). So he needs 17 weeks to have enough money (since after 16 weeks he has \$124, which is not enough, so he needs 17 weeks).

Wait, but maybe the problem is just asking for the division result without rounding up? Let's see the problem statement: "figure out the actual number of weeks". So maybe we can present it as a decimal or a fraction, but usually, in such problems, we round up to the next whole number because you can't have a fraction of a week. But let's do the exact calculation: \( 125\div7.75 = \frac{12500}{775}=\frac{500}{31}\approx16.13 \). So if we consider that he can save part of a week? No, weeks are discrete. So the actual number of weeks needed is 17 (since 16 weeks is not enough). But let's check the estimation again. Estimation: round 7.75 to 8, 125 to 128 (since 128 is divisible by 8, 128÷8=16). So estimation is 16 weeks. Then actual is \( 125\div7.75\approx16.13 \), so approximately 16 or 17 weeks. But the problem says "figure out the actual number of weeks", so we need to do the division.

Answer:

Estimation: Approximately 16 weeks.
Actual number of weeks: \( \frac{125}{7.75}\approx16.13 \), so he needs 17 weeks (since 16 weeks gives \$124, which is less than \$125). If we consider the exact decimal, it's approximately 16.13 weeks, but since weeks are whole, 17 weeks.