# Area of a rectangle

#### Graham2487

##### New member
While helping my son with yr 6 maths homework i stumbled on this question.

A rectangle is cut into two pieces.
The area of A is 5/8 of the area of the rectangle.
The area of A is 28cm2 greater than the area of B.
What is the area of the rectangle ?

Any help with this, a method or formula, would be much appreciated.
Many thanks
Graham

#### tkhunny

##### Moderator
Staff member
Area of Trapezoid might be useful, but you don't really need it.

R = Area of Rectangle = (8/8)*R <== This may seem meaningless or obvious. That's fine.
a = Area of A Portion = (5/8)*R
b = Area of B Portion = a - 28 cm^2 = (3/8)*R <== Were did I get this last piece, (3/8)*R? Why did I SUBTRACT 28 cm^2?

• Otis

#### Graham2487

##### New member
Thanks but yes why and from what did you subtract the 28cm^2 ?

#### Otis

##### Senior Member
… a method or formula …
Hi Graham. It's helpful to know also the topic(s) the student has been studying, for which this exercise was assigned. Do you know?

If your son tried anything, or is confused by anything specific, we'd like to know about that, too. Thanks #### Jomo

##### Elite Member
While helping my son with yr 6 maths homework i stumbled on this question.

A rectangle is cut into two pieces.
The area of A is 5/8 of the area of the rectangle.
The area of A is 28cm2 greater than the area of B.
What is the area of the rectangle ?

Any help with this, a method or formula, would be much appreciated.
Many thanks
Graham
I can only assume that the rectangle was cut into two pieces called A and B. Suppose the area of the complete rectangle is R cm^2

Then the area of part A is (5/8)R cm^2.
Part B must then be the the remaining (3/8) of the rectangle. Then its area is (3/8)R

Since the area of A is more than the area of B before we can say that area A = area B but must 1st add some area to B.

So Area A = Area B + 28cm^2.

You want the area of the entire rectangle which is area A + area B = R.

R = (area B + 28) + area B = ((3/8)R + 28 ) + (3/8)R = (6/8)R + 28 = (3/4)R + 28.

R = (3/4)R + 28.

So 28 must represent (1/4)R. That is (1/4)R = 28. Then R = 112.

Check: area A is (5/8)112=70 cm^2
area B = (3/8)112= 42cm^2
The difference of 70 and 42 is 28!

#### Graham2487

##### New member
I can only assume that the rectangle was cut into two pieces called A and B. Suppose the area of the complete rectangle is R cm^2

Then the area of part A is (5/8)R cm^2.
Part B must then be the the remaining (3/8) of the rectangle. Then its area is (3/8)R

Since the area of A is more than the area of B before we can say that area A = area B but must 1st add some area to B.

So Area A = Area B + 28cm^2.

You want the area of the entire rectangle which is area A + area B = R.

R = (area B + 28) + area B = ((3/8)R + 28 ) + (3/8)R = (6/8)R + 28 = (3/4)R + 28.

R = (3/4)R + 28.

So 28 must represent (1/4)R. That is (1/4)R = 28. Then R = 112.

Check: area A is (5/8)112=70 cm^2
area B = (3/8)112= 42cm^2
The difference of 70 and 42 is 28!
Much appreciated for your help. This will look a bit daunting for my son as im sure he's not learnt sums as complex looking as this just yet. Would there be a simplified version ?? Also how did you get from (3/8)R = (6/8)R ?? I get the 3/4 and therefore 28 is a 1/4 but how did you initially get to the (6/8) ??
Many thanks
Graham

#### Otis

##### Senior Member
… This will look a bit daunting for my son …
Hi Graham. Beginning students need exposure and practice, to understand symbolic math. I'm glad you're there to offer encouragement because symbolic reasoning skills are very important. If your son is currently studying pre-algebra topics, then this is a good opportunity to become familiar with using letters as symbols that represent numbers and using substitution methods to rewrite relationships (equations). We can remove "clutter" from some equations, by leaving out defining words and units.

Let R represent the rectangle's area
Let A represent the area of piece A
Let B represent the area of piece B

With these three symbol definitions, I expect your son understands why we can write:

R = A + B

We're told that:

B = (3/8)R

We're also told that A is 28 more than B. As Jomo explained, that means we can write:

A = B + 28

Symbol A and the expression B+28 both represent the same number (they're equal). Therefore, we're now free to replace symbol A with the expression B+28 anywhere we choose. Let's make that replacement in our equation for the total area:

R = A + B

R = (B + 28) + B

It doesn't matter in what order we add numbers, so we can rewrite the new expression for R:

R = B + B + 28

We've discovered that the rectangle's area is twice the area of piece B plus 28 more. Here's where 6/8 comes from. We have another expression for the area of piece B (the exercise provided it, and it's written above in red). We substitute that expression for symbol B:

R = (3/8)R + (3/8)R + 28

R = (6/8)R + 28

I think you understand the rest:

R = (3/4)R + (1/4)R
R = (3/4)R + 28

Hence, 28 must be 1/4th of R.

28 × 4 = 112

It's good form to answer word problems with a sentence, including units. The rectangle's area is 112 cm2.

The basic strategy was to start with the relationship R=A+B, followed by using given information to substitute expressions for symbols A and B, to obtain an equation that contains only the symbol whose value we're trying to find (R). If your son has specific questions, let us know. We could also help you create a model for this exercise, by cutting paper sheets into pairs of labeled pieces. #### Dr.Peterson

##### Elite Member
While helping my son with yr 6 maths homework i stumbled on this question.

A rectangle is cut into two pieces.
The area of A is 5/8 of the area of the rectangle.
The area of A is 28cm2 greater than the area of B.
What is the area of the rectangle ?
This will look a bit daunting for my son as im sure he's not learnt sums as complex looking as this just yet. Would there be a simplified version ??
It would be very helpful to see what sort of problems your son has been doing, and what methods were used. That way, we could try to stick to methods he's used to.

This can be done easily enough without algebra, if he's not doing much of that yet. Here's how I'd do this with a younger student:

Draw a rectangle, and divide it into two unequal parts. If the larger one is 5/8 of the whole, then the smaller one must be 3/8, right?

So A is 5/8, and B is 3/8, of the whole.

The difference between A and B is supposed to be 28. But the difference of 5/8 and 3/8 is 2/8, or 1/4 of the whole. So 28 must be 1/4 of the whole.

What is 28, 1/4 of? The whole must be 4 times 28, which is 112. So the answer is 112 cm^2.

Does that make sense? Do you think that's more like what he's been learning? This is actually a little harder than the algebra in a sense, because you have to do more special thinking and less routine. But that's good for a student! Thinking hard is good. And this more concrete thinking is good preparation for the abstract thinking of algebra.

Now, when we solve a problem, we need to check that our answer really works. So we work through the problem with our answer:

A rectangle (area 112 cm^2) is cut into two pieces.​
The area of A is 5/8 of the area of the rectangle (5/8 * 112 = 70).​
The area of A is 28 cm^2 greater than the area of B (B is 112 - 70 = 42; and 70 - 42 = 28!).​

So it worked.

#### Jomo

##### Elite Member
Much appreciated for your help. This will look a bit daunting for my son as im sure he's not learnt sums as complex looking as this just yet. Would there be a simplified version ?? Also how did you get from (3/8)R = (6/8)R ?? I get the 3/4 and therefore 28 is a 1/4 but how did you initially get to the (6/8) ??
Many thanks
Graham
(3/8) + (3/8) = 6/8 = 3/4