Mr. Tamer Elsawy - GAT Part 2 Quantitative Review

Vocabulary and problem-solving flashcards derived from Mr. Tamer Elsawy's GAT Part 2 lecture notes, covering quantitative reasoning, algebra, geometry, and arithmetic word problems.

GAT Question 449: Perimeter and Area of a Triangle

If the perimeter of a triangle is $24$ and its sides are $x$, $x+2$, and $x+4$, the area of the triangle is $24$.

(Calculation: $x + (x+2) + (x+4) = 24 \implies 3x + 6 = 24 \implies x = 6$. The side lengths are $6$, $8$, and $10$, forming a right triangle. Area = $\frac{1}{2} \times 6 \times 8 = 24$)

GAT Question 500: Ratio of Shaded Area in Equilateral Triangle

The ratio of the shaded area to the whole equilateral triangle is $1 : 16$.

(The whole equilateral triangle is divided into $4$ medium equilateral triangles, and one medium triangle is further divided into $4$ smaller equilateral triangles, giving $16$ total small units)

GAT Question 501: Angle Traversed by Minute Hand

The angle turned by the minute hand from $4:00\,\text{pm}$ to $7:30\,\text{pm}$ is $1260^\circ$.

(Calculation: Elapsed time = $3.5\,\text{hours}$. Angle = $3.5 \times 360^\circ = 1260^\circ$)

GAT Question 504: Area of an Equilateral Triangle from Perimeter

If the perimeter of an equilateral triangle is $3$, its area is $\frac{\sqrt{3}}{4}$.

(Calculation: Side length $s = \frac{3}{3} = 1$. Area = $\frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4}(1)^2 = \frac{\sqrt{3}}{4}$)

GAT Question 505: Exterior Angle Theorem

In the given triangle with exterior angle $120^\circ$ and non-adjacent interior angle $50^\circ$, the value of $x$ is $70$.

(Calculation: An exterior angle equals the sum of the two opposite interior angles: $x + 50 = 120 \implies x = 70$)

GAT Question 507: Algebraic Difference of Squares

If the sum of two numbers is $25$ and the difference between their squares is $125$, the greatest number is $15$.

(Calculation: $a^2 - b^2 = (a-b)(a+b) = 125 \implies (a-b)(25) = 125 \implies a-b = 5$. Solving $a+b=25$ and $a-b=5$ gives $2a = 30 \implies a = 15$)

GAT Question 510: Quantitative Comparison of Ratios

Given $0 < A < B$ and $B > 1$, the largest term among $\frac{A}{B}$, $\frac{B}{A}$, $\left(\frac{A}{B}\right)^2$, and $\left(\frac{B}{A}\right)^2$ is $\left(\frac{B}{A}\right)^2$.

(Explanation: Since $B > A > 0$, $\frac{B}{A} > 1$. Squaring a number greater than $1$ yields an even larger value)

GAT Question 512: Inscribed Circle and Circumscribed Square

If the area of a circle is $16\pi\,\text{cm}^2$, the area of the square enclosing it is $64\,\text{cm}^2$.

(Calculation: $\pi r^2 = 16\pi \implies r = 4\,\text{cm}$. The diameter $d = 8\,\text{cm}$ equals the side length of the square. Area of square = $8^2 = 64\,\text{cm}^2$)

GAT Question 516: Age Problem

A father's age is $8$ times his son's age. After $19$ years, the son will be $24$. The present age of the father is 40$.\n\n(Calculation: Son's current age = 24 - 19 = 5$. Father's current age =$8 \times 5 = 40$$)

GAT Question 520: Angles from Intersecting Lines

For two intersecting lines forming a top angle of $140^\circ$, the value of $x + y$ is 80$.\n\n(Calculation: Vertically opposite angles give 140^\circ$. Adjacent supplementary angles give$x = 180^\circ - 140^\circ = 40^\circ$and$y = 40^\circ$. Thus,$x + y = 40 + 40 = 80$$)

GAT Question 522: Rhombus Interior Angles

In the rhombus with consecutive interior angles $6x$ and $4x$, the value of $y$ (which is opposite to $4x$) is 72^\circ$.\n\n(Calculation: Consecutive interior angles are supplementary: 6x + 4x = 180^\circ \implies 10x = 180^\circ \implies x = 18^\circ$. Opposite angles in a rhombus are equal:$y = 4x = 4(18^\circ) = 72^\circ$$)

GAT Question 523: Periodic Meeting Least Common Multiple

If three sons visit every $3$ days, $5$ days, and $6$ days respectively, they will meet all together $12$ times in a year.

(Calculation: $\text{LCM}(3, 5, 6) = 30$ days. In a $360$-day year, $\frac{360}{30} = 12$ total combined visits)

GAT Question 526: Modular Sequence Cycle

In the repeating order orange, apple, lemon, and mango, the $135^{\text{th}}$ item is lemon.

(Calculation: The pattern repeats every $4$ items. $135 \div 4 = 33$ with a remainder of $3$, corresponding to the 3rd item in sequence: lemon)

GAT Question 531: Circumference of Circle with Inscribed Right Triangle

The circumference of the circle containing a right triangle with legs $6\,\text{cm}$ and $8\,\text{cm}$ is $31.4\,\text{cm}$.

(Calculation: Hypotenuse = diameter $d = \sqrt{6^2 + 8^2} = 10\,\text{cm}$. Circumference = $\pi d = 3.14 \times 10 = 31.4\,\text{cm}$)

GAT Question 532: Decreasing Pattern Sequence

In the sequence $20, 18, 15, 11, \dots$, the first negative number is -7$.\n\n(Calculation: Successive differences are -2, -3, -4, -5, -6, -7$. Terms:$11 - 5 = 6$,$6 - 6 = 0$,$0 - 7 = -7$$)

GAT Question 535: Sum of Quadratic Equation Roots

The sum of the roots for the equation $x^2 - 1 = 0$ is 0$.\n\n(Calculation: Roots are x = 1$and$x = -1$. Sum =$1 + (-1) = 0$$)

GAT Question 542: Currency Banknotes Word Problem

A man has $120$ banknotes in $5\,\text{SR}$ and $10\,\text{SR}$ denominations. If the number of $10\,\text{SR}$ bills is $5$ times the number of $5\,\text{SR}$ bills, the total amount of money is $1100\,\text{SR}$.

(Calculation: Let $n$ be $5\,\text{SR}$ bills. $n + 5n = 120 \implies 6n = 120 \implies n = 20$. Value = $(20 \times 5) + (100 \times 10) = 100 + 1000 = 1100\,\text{SR}$)

GAT Question 551: Triangle Inequality Test

The side lengths $1, 3, 4$ cannot form a triangle because $1 + 3 = 4$, violating the Triangle Inequality Theorem (the sum of any two side lengths must be strictly greater than the third side length).

GAT Question 555: Circular Division Movement

Starting from point $Q$ on a circle divided into $8$ equal sectors, moving $\frac{4}{8}$ counter-clockwise and then $\frac{1}{4}$ clockwise stops at point $W$.