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Solve the following:
$(a)$ The teacher tells the class that the highest marks obtained by a student in her class is twice the lowest marks plus 7. The highest score is 87. What is the lowest score?
$(b)$ In an isosceles triangle, the base angles are equal. The vertex angle is 40°. What are the base angles of the triangle? [Remember, the sum of three angles of a triangle is $180^{\circ}$].
$(c)$ Sachin scored twice as many runs as Rahul. Together, their runs fell two short of a double century. How many runs did each one score?
To do:
We have to solve the given questions.
Solution:
(a) Highest score $=87$.
Let the lowest marks be $x$
According to question,
Highest marks obtained $= 2\times x+7$
$87=2x+7$
$87-7=2x$
$80=2x$
$x=\frac{80}{2}$
$x=40$
(b) Let one of the base angles be $b$. Since the triangle is isosceles, the other base angle will also be $b$.
Vertex angle $=40^{\circ}$.
The sum of the three angles of a triangle$=180^{\circ}$
$b+b+40^{\circ}=180^{\circ}$
$2b+40^{\circ}=180^{\circ}$
$2b=180^{\circ}-40^{\circ}$
$2b=140^{\circ}$
$b=\frac{140^{\circ}}{2}$
$b=70^{\circ}$
(c) Let Rahul's score be $x$.
This implies,
Score of Sachin $=2\times x$
$=2x$.
According to question,
$x+2x=198$ [1 century$=100$, thus, double century$=200$]
$3x=198$
$x=\frac{198}{3}$
$x=66$
So, Rahul’s score $=66$ runs
And Sachin’s score$=2x$
$=2\times66$
$=132$ runs