T ate 100 cookies in 5 days, each day eating 6 more than...

[rex1973]

Tim ate 100 cookies in 5 days. Each day he ate 6 more than the day before. How many cookies did he eat on the first day ?

I started by taking 100 and subtracting 6 for four days. I do not think this is correct. Can you assist me in another way of looking at this problem?

 

[TchrWill]

Re: problem solving

Tim ate 100 cookies in 5 days. Each day he ate 6 more than the day before. How many cookies did he eat on the first day ?
I started by taking 100 and subtracting 6 for four days. I do not think this is correct. Can you assist me in another way of looking at this problem?

Look at the number of cookies eaten each day as an arithmetic progression, i.e., a + (a + 6), (a + 12), a + 18) and (a + 24)where a = the number eaten on the first day..

The sum total of cookies derives from S = n(a + L)/2 = na + n(n - 1)d/2 where a = the first term, L = the last term, d = the common difference and n = the number of terms.

Having S, n and d, solve for a.

 

[Denis]

Re: problem solving

OR "add 'em up!":
a + (a + 6)+ (a + 12) + (a + 18) + (a + 24) = 100
5a + 60 = 100

 

[soroban]

Hello, rex1973!

Here's a back-door approach . . .

Tim ate 100 cookies in 5 days. Each day he ate 6 more than the day before.
How many cookies did he eat on the first day ?

He ate 100 cookies in 5 days.
. . He averaged 20 cookies per day.

It is as if his eating schedule was: .\(\displaystyle \begin{array}{ccccc} 20 & 20 & 20 & 20 & 20 \end{array}\)

But, of course, he didn't eat the same number every day,
. . the numbers increased by 6 daily.


So we can modify our "schedule" like this:

. . \(\displaystyle \begin{array}{ccccc}20 & 20 & 20 & \;20 & \quad20 \\ \text{-}12 & \text{-}6 & \;0 & \text{+}6 & \text{+}12 \\ \hline 8 & 14 & 20 & \;26 & \quad32 \end{array}\)

Therefore, he ate 8 cookies on the first day.