Question

Please show and explain work :)

3. Four hungry boys went to bed, thinking about the batch of cookies that their mother had just baked. In the middle of the night, each of them got up and ate some of the cookies. Adam got up first. He gave a cookie to the dog, then ate 1/8 of the remaining cookies and went back to bed. Billy was the next to get up, and he ate 1/7 of the cookies. (The dog didn’t get any from him because he thought the dog looked full.) Charlie then got up and gave 3 cookies to the dog. Then he ate 1/5 of the remaining cookies. Daniel was the last to get up, and he ate just 6 cookies. (He offered one to the dog, but the dog just turned away with a decided grimace.) When their mother went into the kitchen the next morning, she found ants, a pale-looking dog, a lot of crumbs, and only a few cookies, so she quickly whipped up another small batch to increase the amount by 3/5. Then she had a total of 48 cookies. How many cookies had she baked the day before?

Answer #1

Let the number of cookies baked by the boys’ mother on the first day be x.

Adam gave 1 cookie to the dog and ate 1/8(x-1) cookies.

Billy ate x/7 cookies.

Charlie gave 3 cookies to the dog and ate 1/5[x–{1+1/8(x-1)+x/7+3}] =1/5(41x/56-31/8) = 41x/280- 31/40 cookies.

Daniel ate 6 cookies.

The total number of cookies eaten by the dog is 4 and the total number of cookies eaten by the 4 boys is 1/8(x-1)+x/7+41x/280-31/40+6 = 116x/280+204/40. Therefore, the number of cookies remaining the next day is x –[4+116x/280+204/40]= 164x/280-364/40= 41x/70-91/10.

Further, the boys’ mother baked 3/5^{th} of
(41x/70-91/10) cookies the next day so the total number of cookies
now is (1+3/5)( 41x/70-91/10) = 8/5(41x/70-91/10) = 164x/175
-364/25.

Thus, 164x/175 -364/25 = 48 or, 164x/175 = 48+364/25 = 1564/25 so that x = (1564/25)*175/164= 62.56. Thus the boys’ mother had baked 63 cookies the day before.

**Note:**

There is something wrong about the data. Most probably, the
word” remaining “ is missing from **Billy was the
next** **to get up, and** **he ate 1/7 of
the cookies. If so, then** Billy ate 1/7[ x –{1+(x-1)/8} =
(x-1)/8 cookies instead of x/7 cookies. Then Charlie ate
1/5[x–{1+1/8(x-1)+(x-1)/8+3}]= 3x/20 - 3/4 cookies. Then the total
number of cookies eaten by the boys is 1/8(x-1) +(x-1)/8 +3x/20 -
¾+6 = 2x/5+5. The dog ate 4 cookies so that the total no. of
cookies eaten is 2x/5 +9. The no. of cookies remaining is x –(2x/5
+9) = 3x/5-9 The boys’ mother baked 3/5^{th} of (3x/5-9)
cookies the next day so the total number of cookies now is (1+3/5)(
3x/5-9) = (8/5)*(3x/5-9) =24x/25- 72/5 = 48 . Hence 24x/25 =
48+72/5= 312/5. Then x = (312/5)*(25/24) =13* 5 = 65. This would be
the correct answer.

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