complementary angles

Question

The ratio of the measures of two complementary angles is 4 to 5. The smaller measure is increased by 10%. By what percent must the larger be decreased so that the two angles remain complementary? 

Answer 1

4 to 5 is the ratio

4+5=9

90/9=10

40º and 50º are the angles

10% of 40 is 4

40 becomes 44

90-44=46

50 decreased to 46

50-46=4

4/50=8/100=8% decrease

Answer 2

Complementary angles add to 90 degrees.

 

Let smaller angle = x

Let larger angle = y

 

Use these variables, we can get equations to represent the ratio and sum of angle.

 

x / y = 4 / 5

 

x = (4/5)y         eq1   ----> ratio

 

x + y = 90         eq2

 

 

Substitute eq1 into eq2.

 

(4/5)y + y = 90

 

 

Solve for y.  Multiply both sides of the equation by 5.

 

4y + 5y = 450

 

9y = 450

 

  y = 50

 

Then  x = 40

 

 

These are your original angles: 40 and 50. 

 

Now, we make the changes to these angles.  The smaller angle is increased by 10%.

 

40(1 + 0.10) + 50(1 - p) = 90

 

 

Solve for to get your percent.

 

40(1.90) + 50 - 50p = 90

 

76 + 50 - 50p = 90

 

126 - 50p = 90

 

     -50p = -36

 

         p = 0.72

 

The larger angle must decreased by 72% in order for the two angles to remain complementary.

 

New smaller angle = 40 x 1.90 = 76

New larger angle = 50 x 0.28 = 14

 

Answer 3

The two angles are 40° and 50°.

Increasing the small by 10% makes it 44°.

The larger angle is decreased by 4/50 or 8%.