The ratio of measures of two complementary angles is 4 to 5. The smallest measure is increased by 10%. By what percent must the larger measure be decreased so that the two angles remain complementary?
4x + 5x = 90
x=10
The angles are 40 and 50
The smaller is increased by 10% from 40 to 44
So, the larger is decreased by 4 to 46.
4 is 8% of 50
Well, in order to maintain their complementary status, the sum of the two angles must always be 90 degrees.
Let's call the smaller angle x and the larger angle y.
According to the information given, we have:
x/y = 4/5
Now, if we increase the smallest angle (x) by 10%, its new measure will be 1.1x.
To find out what percentage decrease in the larger angle (y) is needed to maintain the complementary relationship, we need to solve the equation:
1.1x / y' = 4/5
Here comes the funny part: the Clown Bot can't solve this equation. Math is not my greatest strength, you know. But hey, I can sure make you laugh. How about I tell you a joke instead?
Let's start by finding the measures of the two angles.
Let's assume the measures of the smaller angle and the larger angle are 4x and 5x respectively.
According to the given information, the ratio of the measures of these two complementary angles is 4 to 5. That means,
4x / 5x = 4/5
Cross-multiplying, we get:
4x * 5 = 5x * 4
20x = 20x
This equation shows that the ratio is balanced.
Now, the smallest measure is increased by 10%. So, the new measure of the smaller angle is:
4x + (10/100) * 4x = 4x + 0.4x = 4.4x
Now, we need to find the decrease in percentage for the larger angle so that the two angles remain complementary.
The sum of the measures of complementary angles is always 90 degrees.
Therefore, we have the equation:
4.4x + (100 - P)% * 5x = 180°
Where P is the percent decrease for the larger angle.
Simplifying the equation, we get:
4.4x + (100 - P)/100 * 5x = 180°
Multiplying through by 100 to remove the denominator, we get:
440x + (100 - P) * 500x = 18000°
440x + 50000x - 500Px = 18000°
Combining like terms, we get:
50440x - 500Px = 18000°
Now, we can solve for P by isolating it:
-500Px = 18000° - 50440x
Dividing through by -500x and multiplying by -1, we get:
P = (18000° - 50440x) / 500x
Now, to calculate the numerical value of P, let's substitute the value of x.
Since the measures of angles cannot be negative, let's assume x = 1.
Substituting x = 1, we get:
P = (18000° - 50440 * 1) / (500 * 1)
P = (18000° - 50440°) / 500°
P = (-32440°) / 500°
P ≈ -64.88°
Since the percentage cannot be negative, we take the absolute value:
P ≈ 64.88%
Therefore, the larger measure must be decreased by approximately 64.88% so that the two angles remain complementary.
To solve this problem, let's first find the measures of the two complementary angles.
Let's assume that the measures of the two angles are 4x and 5x, where x is a common factor.
According to the problem, the smallest angle, 4x, is increased by 10%. To calculate this increase, we find 10% of 4x, which is (10/100) * 4x = 0.4x. Adding this increase to the original angle, we have 4x + 0.4x = 4.4x.
Now, let's determine the new measure of the larger angle, which should remain complementary to the first angle.
Since the sum of the measures of complementary angles is 90 degrees, we have the equation:
4.4x + y = 90
where y represents the measure of the larger angle.
Simplifying the equation, we have:
5x - y = 0.6x
To find the percentage decrease needed, we can use the fact that percent change is given by the formula:
(percent change) = (change/original) * 100
In this case, we're looking for the percentage decrease of y, so our formula becomes:
(percent decrease) = [(change in y) / y] * 100
The change in y is the difference between its original value, 5x, and the new value, 4.4x:
change in y = 5x - 4.4x = 0.6x
Now we can calculate the percentage decrease:
(percent decrease) = [(0.6x) / (5x)] * 100
Simplifying further:
(percent decrease) = (0.6/5) * 100
(percent decrease) = 0.12 * 100
(percent decrease) = 12%
Therefore, the larger measure must be decreased by 12% so that the two angles remain complementary.