# There are 14 desks of 4 types; desks with 1 drawer, 2 drawers, 3 drawers and 4 drawers. There are 33 drawers altogether.The number of 2 drawer desks and 3 drawer desks combined is the same as the...

There are 14 desks of 4 types; desks with 1 drawer, 2 drawers, 3 drawers and 4 drawers. There are 33 drawers altogether.

The number of 2 drawer desks and 3 drawer desks combined is the same as the number of 1 drawer desks. How many 1 drawer desks are there?

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### 1 Answer

This problem can be solved using systems and substitution.

Assign variables as follows:

a: 1 drawer desks

b: 2 drawer desks

c: 3 drawer desks

d: 4 drawer desks

Here is what we know:

a + b + c + d = 14 (because there are 14 desks)

1a + 2b + 3c + 4d = 33 (because there are 33 drawers)

b + c = a (because the combination of 2 and 3 drawer desks equals the number of 1 drawer desks)

Substitute (b + c) in for a in both equations.

(b + c) + b + c + d = 14

2b + 2c + d = 14

1(b + c) + 2b + 3c + 4d = 33

3b + 4c + 4d = 33

You now have the following system of equations:

2b + 2c + d = 14

3b + 4c + 4d = 33

Multiplying the first equation by -4.

-8d + -8c + -4d = -56

3b + 4c + 4d = 33

Now add the equations. The variable d has been eliminated.

-5b + -4c = -23

To make this equation easier to work with, multiply through by -1.

5b + 4c = 23

Solve for b.

b = (23 - 4c) / 5

Since b represents 2 drawer desks, it must be a whole number. The only value of c that will produce a whole number answer is 2.

b = (23 - 4 * 2) / 5

b = (23 - 8) / 5

b = 15 / 5

b = 3

So now we know that b = 3 and c = 2.

We already know that b + c = a.

Therefore...

b + c = a

3 + 2 = 5

a = 5

We also know that there are 14 desks. Therefore...

a + b + c + d = 14

5 + 3 + 2 + d = 14

10 + d = 14

d = 4

**There are 5 desks with one drawer, 3 desks with two drawers, 2 desks with three drawers, and 4 desks with four drawers.**

You can check this using the second equation.

1a + 2b + 3c + 4d = 33

1(5) + 2(3) + 3(2) + 4(4) = 33

5 + 6 + 6 + 16 = 33

33 = 33

This answer works.

**Answer: There are 5 desks with one drawer.**