A puzzling math question....?

Question

A piece of paper 18 cm x 24 cm has to be shrunk proportionally to 4/9 of its area size.

What will be the length and width of the piece of paper after its area has been reduced?

All calculated so far is this:

18 cm X 24 cm = 432 cm² (old area)

4/9 X 432 cm² = 192 cm² (new area)

192 cm² Z cm
------------- = ------------
432 cm² 18 cm

Z cm = 8 cm (width)


I must be doing something really wrong cuz apparently the width = 12 cm and the length = 16 cm.

And Ive just been goin over this a ton of times...but i cant seem to figure it out.

Any ideas? =)

You guys are awesome! Thank you sooooo much! I wish i could pick you all for best answer. =))

Answer 1

Let z be the amount to multiply each dimension by:
Let A be the original area
Let A' be the new area

A = W * L
A' = zW * zL
A' = z²(W * L)
A' = z²A

We want the new area (A') to be 4/9 the old area (A).

That means z² = 4/9
And therefore:
z = sqrt(4/9)
z = 2/3

If the original dimensions are 18 cm * 24 cm. The new dimensions will each be 2/3 of that (12 cm * 16 cm).

The old area will be 432 cm², the new are will be 192 cm² which is 4/9 the original.

Thus the new dimensions should be 2/3 the old or 12 cm * 16 cm.

Answer 2

12 x 16.


First, the area is 18 x 24 =432

432 x 4/9 =1728/9 = 192 (area of new rectangle)

192 sq cm in an rectangle having the sides proportionate to 3 x 4 (same as 18 x 24) is 12 x 16.

Check the work: 12 x 16 = 192 = 432 x 4/9

Answer 3

In order to solve this problem, you need to figure out how much to shrink the length and width of the piece of paper. Since the units of area are going to be squared, in order to deal with the length and width induvidally, you will have to take the square root of the units. You would need to do this to the 4/9 also to find out how much to shrink each side. The square root of 4/9 is 2/3. 2/3 of 18cm is 12 cm. 2/3 of 24 cm is 16 cm. The shrunken area is now 12 cm * 16 cm = 288 cm^2. 192 cm^2 is 4/9 of 432 cm^2

Answer 4

Area is squared measure. L & W are linear so they change by a factor of √(4/9)=2/3

there for the width will be 18*√4/9)=18*(2/3)=12
the length will be 24*2/3=16

the smaller rectangle will be 12X16

check
large A=18*24=432
small A=12*16=192
432*4/9=192

Answer 5

18 cm X 24 cm = 432 cm² (old area)

4/9 X 432 cm² = 192 cm² (new area)

18 / 24 = 0.75 (a ratio)

0.75x^2 = 192

x^2 = 192/0.75 = 256

x = 16

0.75x = 12

12 X 16 = 192

new dimensions are 12 cm X 16 cm

Answer 6

4/9 of the old area is 192 cm2. The ratio of width to height is 3/4 and that has to hold true after you shrink.

So (3/4x)*x=192 and solve for x which will be the height.

I got 12 x 16 cm.

Answer 7

You're on the right track. You've already got that the new area needs to be 192cm. Let's set up the problem
Side two, the longer of our sides is 24 cm. Let's call that Y.
Side one, the shorter of our sides, is 18cm. Lets make that x.

y=24, x =18. If we set the problem up so that we work in terms of y, then x=3/4y.

Area is x*y
3/4y*y=492
solve for y would give 24cm, so our basic formula is correct.

Because we want a proportional piece of paper, we use the same formula to calculate the new length and width.

3/4y * y = 192
3/4y^2=192
y^2=256
y=16
Then, find x
x=3/4y = 12

Does that make sense? I'm sorry if my explanation isn't the greatest!

Answer 8

18 cm X 24 cm = 432 cm² (old area)

4/9 X 432 cm² = 192 cm² (new area)

You are right up to now.

Let solve it using similarity Method:
L.S.F = 18 : 24 (ratio of width to length)
= 3 : 4

A.S.F = LSF²
= 9 :16

I think you will prefer to continue yourselves

If you have problem email me

Answer 9

(4/9) * 432 = 192 sq. cm ... correct
width*length = 192 sq. cm (new equation #1)
w*l = 192 sq. cm (just reducing variable names to 1 letter)
(w/l) = (18/24) = 0.75 (new equation #2 ... proportionality)

So: (1) w*l = 192 and (2) (w/l)=0.75
Working with (2): w/l = 0.75 then w=0.75*l

Working with #1 (substitution): w*l=192 then (0.75*l)*l=192
0.75*l*l = 192
l*l = 256
l = length = 16
w=0.75*l = 0.75*16 = 12

Solution: w=12cm, l=16cm

Answer 10

Let x & y be the new length & width of the paper after reduction.

x * y = 192

y/x = 18/24 = ¾

y = ¾x

x * ¾x = 192
¾x² = 192
x² = 256
x = 16cm
y = 12cm