Question

John currently has $500 in his lunch account. He spent $15 each day on lunch. In how many days will John have $245?

Answer 1

18 days.

Explanation:

Step 1) Building an arithmetic sequence
We can build an arithmetic series with the given information.
The
`n`th term of an arithmetic series is given by the formula
`a_n = a_1 + (n - 1)d` .

`a_1 = 500\\d = -15\\\to a_n = 500 - 15(n - 1) = 515 - 15n`

Step 2) Finding the n for which a(n) = 245

As this step's title implies, we need to solve the equation
`a_n = 245`:

`515 - 15n = 245 \text{ / Subtract 245}\\515 - 15n - 245 = 245 - 245\\\to 270 - 15n = 0 \text{ / Add 15n}\\270 - 15n + 15n= 0 + 15n\\\to 270 = 15n \text{ / Divide by 15}\\(270)/(15) = (15n)/(15)\\\to 18 = n`

John will have $245 in 18 days.

Answer 2

17 days

Explanation:

Since John starts with $500 and each day, he’ll spend $15 on lunch. This means that:

• On his first day, he’ll spend $15.
• On his second day, he’ll spend another $15 which is total spend of $30 ($15 x 2)
• On his third day, he’ll spend total of $45 ($15 x 3)

Therefore, on his nth day, he’ll spend $15n where n represents the day. The problem can be modeled into an equation of
`\displaystyle{500-15n=245}` since we will be finding how many days it will take to reach $245.

Then solve for n-term which we determines as the day variable:

`\displaystyle{500-15n-245=0}\\\\\displaystyle{255-15n=0}\\\\\displaystyle{255=15n}\\\\\displaystyle{n=17}`

Therefore, on his 17th day, he’ll have $245. Hence, the answer is 17 days.