After two successive increments, Gopal’s salary became 187.5% of his initial salary. If the percentage of salary increase in the second increment was twice of that in the first increment, then the percentage of salary increase in the first increment was ?
1) 25
2) 20
3) 27.5
4) 30
Solution
Using Options
1) 25 => 1.25 x 1.5 (second increment double of first) = 1.875 Answer
2) 20 => 1.2 x 1.4 = 1.68 XX
3) 27.5 => 1.275 x 1.55 = 1.976 XX
4) 30 => 1.3 x 1.6 = 2.08 xxx
A regular octagon ABCDEFGH has sides of length 6 cm each. Then the area, in sq. cm, of the square ACEG is
1) 36(2 + √2) 2) 36(1 + √2) 3) 72(2 + √2) 4) 72(1 + √2)
Solution
If AH is 6 than AG visually looks around 10 or 11. therefore the area of square should be around 100 to 121
Calculated Values:
\( 1.\quad 36(2 + \sqrt{2}) \approx 122.91 \) Correct Answer
\( 2.\quad 36(1 + \sqrt{2}) \approx 86.91 \)
\( 3.\quad 72(2 + \sqrt{2}) \approx 245.82 \)
\( 4.\quad 72(1 + \sqrt{2}) \approx 173.82 \)
The midpoints of sides AB, BC, and AC in ΔABC are M, N, and P, respectively. The medians drawn from A, B, and C intersect the line segments MP, MN and NP at X, Y, and Z, respectively. If the area of ΔABC is 1440 sq cm, then the area, in sq cm, of ΔXYZ is
Solution
The points M, N, and P are the midpoints of sides AB, BC, and AC of triangle \( \triangle ABC \). Therefore, triangle \( \triangle MNP \) is the medial triangle.
All 4 Triangles look similar therefore area will be 1/4, ie 1/4 of 1440 = 360 and 1/4 of 360 = 90.
It is a known geometric property that the area of the medial triangle is one-fourth that of the original triangle. So, \[ \text{Area}_{\triangle MNP} = \frac{1}{4} \cdot \text{Area}_{\triangle ABC} = \frac{1}{4} \cdot 1440 = 360 \, \text{cm}^2 \]
Now, medians from vertices A, B, and C intersect the opposite sides of the medial triangle at points X, Y, and Z. These intersecting points lie in such a way that triangle \( \triangle XYZ \) is formed within triangle \( \triangle MNP \).
It is a known fact that if medians from vertices of a triangle intersect the opposite sides of the medial triangle, then the triangle formed by those intersection points has one-fourth the area of the medial triangle: \[ \text{Area}_{\triangle XYZ} = \frac{1}{4} \cdot \text{Area}_{\triangle MNP} = \frac{1}{4} \cdot 360 = \boxed{90 \, \text{cm}^2} \]
Final Answer:
\[ \boxed{90 \, \text{cm}^2} \]
