Right angle triangle simple problem

Question

The number of degrees in one acute angle of a right-angled triangle is equal to the number of grades in the other; express both the angles in degrees.

So I have found the following answers :

810/17=47,05... degrees and 810/17=47,05... grades which gives 42,35... degrees

Now, the real answer is the following :

900/19=47,3... degrees and 900/19=47,3... grades which gives 42,63... degrees

The only problem with my answer is the following :

810/17degrees=900/17grades

so : 900/17 grades+ 810/17 grades = 100,5.. grades (but for the rest, everything is fine, I get get 90 degrees perfectly and respect all conditions.)

Would you count this as an error ???

Here's what I did : (Help me see the error)

x degrees= (x+x/9) grades

x grades= (x-x/9) degrees

so..

x degrees+(x-x/9)degrees=90 degrees

17x/9 degrees=90 degrees

17x=810

x=810/17 degrees

By the formulaiton of the problem, we also have 810/17 grades

Conversion

810/17 degrees=(810/17+(810/17)/9) grades=900/17 grades

and

810/17 grades=(810/17-(810/17)/9) degrees= 720/17 degrees

Can somebody tell me where I went wrong ??? (By the way, I see how to obtain the "real" answer, but I don't see why mine would be wrong ...)

Thank you !

Answer 1

The angles of a triangle add up to $180$ degrees, or $200$ grades.

So $0.9 G = D$, where $D$ and $G$ are the measures of an angle in degrees and grades, respectively.

One angle is a right angle, so the other two add up to $90$ degrees, or $100$ grades. Further, the measure of one in degrees (expressed by $D_i$) is the measure of the other in grades (expressed by $G_i$).

So $G_1 + G_2 = 100$, and $G_1 = D_2 = 0.9 G_2,$ so $1.9 G_2 = 100$ or $G_2 \approx 52.63$ grades. Then $G_1 \approx 47.37$ grades.

Multiply both by $0.9$ to get $(D_1, D_2) \approx (47.37, 42.63).$

(Where you went off a bit was in your definition of the conversion from degrees to grades. You have $D = (10/9) G$ and $G = (8/9) D$, neither of which is right. It should be $0.9 G = D$ or $(0.9)^{-1} D = G$.)

Answer 2

I don't understand all the reasoning of your solution, but you made an error when you changed from $x$ degrees = $x+\frac x9$ grades, which is correct, to $x$ grades = $x - \frac x9$ degrees. The first says the number of degrees is $\frac 9{10}$ the number of grades. The second says the number of grades is $\frac 98$ the number of degrees instead of $\frac {10}9$

And yes, if the number of grades adds up to $100.5$ instead of $100$ that is an error. It should add to exactly $100$.