# Thread: Big and small plates at a party... (maybe multiples or factors.... i'm not sure)

1. ## Big and small plates at a party... (maybe multiples or factors.... i'm not sure)

If anyone, could help me with these I m confused. Thanks!

Some fairy cakes are delivered for Janey's party.

Janey can use big plates or small plates to put the cakes on.

If she uses big plates she can put 7 cakes on each plate with 5 left over.

If she uses small plates she can put 5 cakes on each plate with 3 left over.

1) What is the smallest number of cakes that were delivered for Janey's party?

The smallest number of cakes are delivered for Janey's party

2) If she only used big plates for the cakes, how many big plates would she need?

3) If she only used small plates for the cakes, how many small plates would she need?

2. ## Re: Big and small plates at a party... (maybe multiples or factors.... i'm not sure)

Originally Posted by Natasha1
If anyone, could help me with these I m confused. Thanks!

Some fairy cakes are delivered for Janey's party.

Janey can use big plates or small plates to put the cakes on.

If she uses big plates she can put 7 cakes on each plate with 5 left over.

If she uses small plates she can put 5 cakes on each plate with 3 left over.

1) What is the smallest number of cakes that were delivered for Janey's party?

The smallest number of cakes are delivered for Janey's party

2) If she only used big plates for the cakes, how many big plates would she need?

3) If she only used small plates for the cakes, how many small plates would she need?
Have you tried anything? Do you have work you can show?

You want to find the smallest number n such that (n mod 7) = 5, and that (n mod 5) = 3. Do you understand why?

so n = 7k + 5, and n = 5j + 3, and thus 7k + 2 = 5j

see if you can figure out j and k, and thus n, and then finish the problem.

3. ## Re: Big and small plates at a party... (maybe multiples or factors.... i'm not sure)

I hadn't done any work no as I did not understand what to do... But that's kind of helped.

1) So the smallest number of cakes is 33 because 7 x 4 + 5 = 33 and 5 x 6 + 3 = 33

2) The smallest number of large plates is 4 because 7 x 4 + 5 = 33

3) The smallest number of small plates is 6 because 5 x 6 + 3 = 33

4. ## Re: Big and small plates at a party... (maybe multiples or factors.... i'm not sure)

Originally Posted by Natasha1
I hadn't done any work no as I did not understand what to do... But that's kind of helped.

1) So the smallest number of cakes is 33 because 7 x 4 + 5 = 33 and 5 x 6 + 3 = 33

2) The smallest number of large plates is 4 because 7 x 4 + 5 = 33

3) The smallest number of small plates is 6 because 5 x 6 + 3 = 33
33 cakes is the correct answer. But for (2) and (3) you need to find the number of plates she has to use so that all the cakes are on a plate, even if one plate isn't full.

So what does that do to the answers you figured out for (2) and (3) ?

5. ## Re: Big and small plates at a party... (maybe multiples or factors.... i'm not sure)

Hello, Natasha1!

Some fairy cakes are delivered for Janey's party.
Janey can use big plates or small plates to put the cakes on.
If she uses big plates, she can put 7 cakes on each plate with 5 left over.
If she uses small plates, she can put 5 cakes on each plate with 3 left over.

1) What is the smallest number of cakes that were delivered for Janey's party?

Let $N$ = number of cakes.

"If she uses big plates, she can put 7 cakes on each plate with 5 left over."
. . Hence: . $N \:=\:7a + 5\,\text{ for some integer }a.$ .[1]

"If she uses small plates, she can put 5 cakes on each plate with 3 left over."
. . Hence: . $N \:=\:5b + 3\,\text{ for some integer }b.$ .[2]

Equate [2] and [1]: . $5b+3 \:=\:7a+5 \quad\Rightarrow\quad b \:=\:\frac{7a+2}{5}$
. . Hence: . $b \:=\:a + \frac{2(a+1)}{5}$

Since $b$ is an integer, $2(a+1)$ must be a multiple of 5.
The first time this happens is $a = 4.$

Substitute into [1]: . $N \:=\:7(4)+5 \:=\:33$

6. ## Re: Big and small plates at a party... (maybe multiples or factors.... i'm not sure)

Oh I see, so if she has 33 cakes...

2) Then she would need 5 large plates so that she could hold all 33 cakes because 7 x 5 = 35 and have room for a further 2.

3) Then she would need 7 small plates so that she could hold all 33 cakes because 5 x 7 = 35 and have room for a further 2.

Is this correct for questions 2 and 3?

yep