Question posted by Uma from the United States:

Grade/Level: Grade 4

Question: janet bought a box of cookies for her friends. If she
gives each of them 4 cookies, she will have 4 cookies left. if she gives each of them 5 cookies , she will need 5 more cookies. How many cookies are there in the box?

Step 1: Draw 1 long bar to represent the total number of cookies Janet has. Mark out 1 small box to the left to show the 4 cookies given to each of her friends using the 1st way of sharing the cookies.

Step 2: Draw an arrow to show that this way of sharing will be repeated until all the cookies are exhausted. The "?" is to show that we do not know yet how many boxes of "4" will we get.

Step 3: Similarly, we draw another long bar below the first to show that the same number of cookies can be shared in a different manner (5 cookies per child).

Step 4: Draw a dotted line towards the end of the 1st bar to show that she will get 4 cookies left after sharing the cookies with all her friends. Since this 4 cookies are leftovers, they are found inside the original long bar using the 1st way of sharing.

Step 5: Draw a dotted box towards after the end of the 2nd bar to show that she be short of 5 cookies after sharing the cookies with all her friends. Since this 5 cookies are shortfalls, they are found outside the original long bar using the 2nd way of sharing.

Step 6: Draw a bracket below the model to show that she will need 9 extra cookies in the 2nd way of sharing.

From the model,

difference between the first and second way of sharing

= 5 cookies - 4 cookies

= 1 cookie per kid

By giving 5 cookies each, every kid receive 1 extra cookie compared to the 1st way of sharing (4 cookies).

The 4 cookies left from the 1st way of sharing must be given to the kids as the extra cookie per kid (4 kids will benefit by getting 1 extra cookies)

At the same time, she is short of 5 cookies to the remaining kids who have yet to received 1 extra cookie. So, there must be 5 more kids left crying for 1 extra cookies (and shouting "It's not fair the other 4 kids get 1 extra cookie each!" ;p).

Thus, 4 cookies + 5 cookies = 9 extra cookies needed

Since she needs 9 extra cookies,

9 extra cookies / 1 extra cookie per kid = 9 kids

Therefore, there must be 9 kids sharing the cookies.

(9 kids x 4 cookies) + 4 cookies = 40 cookies

Hence, there are 40 cookies in all.