What is the Value of a + b + c + d? GMAT Problem Solving

Question: What is the value of a + b + c + d?

1. 145
2. 180
3. 215
4. 290
5. 315

Correct Answer: D
Solution and Explanation:
Approach Solution 1:

The problem statement portrays an image of three lines crossing each other at one point:

Given:

• The image comprises six angles.
• One angle is 35 degrees.

Asked:

• Find the sum of four angles a, b, c and d.

Let’s assume that the angle which is not marked is equal to the angle marked 35 degrees since they are exactly opposite to each other.
This is because as per the rules of mathematics, the exact opposite angles are equal to each other.
If we make an imaginary curvy line across all the angles, then we obtain a circle.
Therefore, according to the rules of mathematics, we get:
The sum of the angles in the circle is 360 degrees.
Thus the sum of the angle a, angle b, angle c, 35 degrees and the other angle will be equal to 360 degrees.

Therefore, we can say:
Sum of all the angles = 360 degrees
Or, 360 = a + b + c + d + 35 + 35
Or, 360 = a + b + c + d + 70
Or, a + b + c + d = 290
Therefore, the sum of four angles a, b, c and d = 290.

Approach Solution 2:

The problem statement portrays an image of three lines crossing each other at one point:

Given:

• The image comprises six angles.
• One angle is 35 degrees.

Asked:

• Find the sum of four angles a, b, c and d.

Since one angle is given that is equal to 35°.
Therefore we can say, the unmarked angle opposite to this angle is also 35°. Since as per the rules of mathematics, angles vertically opposite to each other are always equal to each other.

As per the diagram it can be seen that there are four 90-degree angles.
We can derive the following from the diagram:
Angle b = 90 – 35 = 55 degrees
Angle a = Angle c = 90 degrees
Angle d + Angle c + 35 = 180 degrees (the sum of three angles of a straight line is 180° as per the rules of mathematics)
This implies, d = 180 – 35 – 90
= 180 – 125
= 55
Therefore, summing up we get:
a + b + c + d = 90 + 55 + 90 + 55 = 290
Hence, the sum of four angles a, b, c and d = 290.

Approach Solution 3:

The problem statement portrays an image of three lines crossing each other at one point:

Given:

• The image comprises six angles.
• One angle is 35 degrees.

Asked:

• Find the sum of four angles a, b, c and d.

According to the rules of mathematics, we know that two angles that lie vertically opposite to each other are always equal to each other.
Therefore, we can say:
Since one angle as given in the diagram is 35 degrees, the unmarked angle opposite it is also 35 degrees. Therefore, the sum of these angles is 35 + 35 =70
Now by analysing the diagram we can say:
Angle a = angle c = 90 degrees.
Therefore, angle a + angle c = 180 degrees.
If we make an imaginary curvy line across all the angles, then we obtain a circle.
Therefore, according to the rules of mathematics, we get:
The sum of the angles in the circle is 360 degrees.
So, angle b + angle d = 360 - 180 - 70 = 110 degrees.
Hence, angle a + angle b + angle c + angle d = 110 + 180 = 290 degrees.

Therefore, the sum of four angles a, b, c and d = 290.

“What is the value of a + b + c + d?”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “GMAT Official Guide Quantitative Review 2018”. The candidates can follow these types of GMAT Problem Solving questions to enhance their mathematical knowledge to crack numerical problems. The candidates can practice more varieties of questions from the GMAT Quant practice papers to score better in the GMAT exam.

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