Word Problems and Applications

Correct answer: A. x² = 5x + 6

The governing skill is translating a verbal relationship into an algebraic equation. If the number is x, its square is x², and five times the number is 5x. The phrase “6 more than 5 times the number” means that 6 is added to 5x, giving 5x + 6. The statement says that the square is equal to this entire quantity, so the required equation is x² = 5x + 6. Option B changes the relationship by adding 5x to the square and equating it to 6. Option C incorrectly makes 5 the coefficient of x², while option D interchanges the roles of 5 and 6. Therefore, option A is the only correct translation.

Correct answer: B. (8)

Let the smaller integer be (x), then (x(x+1)=72) gives (x=8). In exams, write consecutive numbers as (x) and (x+1).

Correct answer: C. (12)

The direct answer is option C: 12. Let the smaller positive integer be x. Since the integers are consecutive, the other one is x+1. Their product gives x(x+1)=132, so x^2+x−132=0. Factoring gives (x−11)(x+12)=0. Hence x=11 or x=−12. Because the question asks for positive integers, x=11 is accepted and −12 is rejected. The two numbers are 11 and 12, so the larger is 12. Option A, 10, is wrong because 10×11=110. Option B, 11, is the smaller number, not the larger one. Option C is correct because 11×12=132 and 12 is larger. Option D, 13, is wrong because 12×13=156. The common mistake is to find the smaller integer and stop. Always check both the product and which number the question asks for.

Correct answer: B. (7) cm

Taking breadth as (x), (x(x+3)=70), which gives (x=7). In area questions, do not forget length times breadth.

Correct answer: A. (6x=96)

Area (=) length (\times) breadth, so (6x=96) is correct. In every word problem, first identify the formula.

Correct answer: B. \(13\,\text{m}\)

For a square, \(\text{area}=(\text{side})^2\). So \((\text{side})^2=169\). Taking square root, \(\text{side}=\sqrt{169}=13\,\text{m}\).

Correct answer: C. (60) m

The correct answer is option C: 60 m. Let the side of the square be s metres. The area of a square is s^2, so s^2=225. Since a length is positive, s=√225=15 m, not -15 m. The perimeter of a square is the sum of four equal sides: P=4s=4×15=60 m. Option A, 30 m, is twice the side and may result from using the wrong formula. Option B, 45 m, does not represent the perimeter of a square with side 15 m. Option C is correct because 15+15+15+15=60. Option D, 75 m, is also not obtained from the square-perimeter formula. Keep area and perimeter separate: area is measured in square metres, while perimeter is measured in metres. Exam cue: from area find one side using a square root, then multiply by 4.

Correct answer: B. (7)

If the number is (x), then (x(x+1)=56), giving (x=7). In the question, next integer means (x+1).

Correct answer: C. (8)

The equation is (x^2=5x+24), or (x^2-5x-24=0), whose positive solution is (x=8). If a positive number is asked, ignore the negative root.

Correct answer: C. (9)

(x^2=4x+45) gives (x^2-4x-45=0), and the positive solution is (9). In such questions, convert the sentence directly into an equation.

Correct answer: B. 8

This is a quadratic word problem based on the sum and product of two numbers. Let one number be x; then the other is 17 − x because their sum is 17. Using the product condition, x(17 − x) = 72. On rearranging, 17x − x² = 72, so x² − 17x + 72 = 0. Factoring gives (x − 8)(x − 9) = 0, so the two numbers are 8 and 9. Since the question asks for the smaller number, the answer is 8, which is option B. We can verify both conditions directly: 8 + 9 = 17 and 8 × 9 = 72. Option C is the larger number, while 6 and 12 do not satisfy both conditions because their sum is 18 and their product is 72.

Correct answer: B. (10)

Putting the numbers as (x) and (19-x), (x(19-x)=90). The solutions are (9) and (10), so the larger number is (10).

Correct answer: B. (12) m

If breadth is (x), then (x(x+5)=84), giving (x=7) and length (12). Check length and breadth separately at the end.

Correct answer: B. 8 cm

The governing concept is the area formula for a rectangle, A = length × breadth, combined with a quadratic equation. Let the breadth be x cm. Then the length is x + 4 cm, so the area condition becomes x(x + 4) = 96. Therefore x² + 4x − 96 = 0. Factoring gives (x + 12)(x − 8) = 0, so x = 8 or x = −12. A length cannot be negative, so the meaningful breadth is x = 8 cm. The corresponding length is 12 cm, and 8 × 12 = 96 confirms the result. Hence option B is correct. The value 6 would give area 60, 10 would give 140, and 12 would give 192 when the length is four centimetres greater, so those distractors fail the area condition.

Correct answer: B. (10) m

The correct answer is option B: 10 m. Let the breadth be x metres. The length is 2 m more, so length=x+2. Area of a rectangle is length×breadth, giving x(x+2)=120. Thus x^2+2x-120=0. Factorising, (x+12)(x-10)=0, so x=-12 or x=10. A physical breadth cannot be negative, so x=10 m. The length is then 12 m and the check is 10×12=120. Option A, 8 m, gives 8×10=80, not 120. Option B is correct. Option C, 12 m, is the length, not the breadth. Option D, 14 m, would give length 16 m and area 224 m², so it fails. The negative algebraic root is rejected because dimensions represent actual positive lengths. Memory cue: define the smaller dimension first, form product=area, solve, then reject negative length.

Correct answer: C. 15 cm

Use the rectangle area relationship, area = length × breadth. Let the breadth be x cm. Since the length is 5 cm greater, it is x + 5 cm. The given area produces x(x + 5) = 150, or x² + 5x − 150 = 0. Factoring gives (x + 15)(x − 10) = 0. The algebraic solutions are x = −15 and x = 10, but a physical breadth must be positive, so the breadth is 10 cm. Consequently, the length is x + 5 = 15 cm. Verification gives 10 × 15 = 150, so option C is correct. Option A is the breadth rather than the requested length; 12 and 20 do not form a pair whose product is 150 and whose difference is 5.

Correct answer: C. (5)

By Pythagoras, (x^2+(x+7)^2=13^2), giving (x=5). In a right triangle, always take the hypotenuse as the largest side.

Correct answer: B. 3 cm

By the Pythagorean theorem, \(x^2+(x+1)^2=25\). This gives \(2x^2+2x-24=0\), or \(x^2+x-12=0\), which factors as \((x-3)(x+4)=0\). Since a side length cannot be negative, \(x=3\) cm, so the shorter side is 3 cm. The value 4 cm is the other perpendicular side, not the shorter one. Exam tip: verify the result using the familiar \(3,4,5\) right-triangle relationship.

Correct answer: C. 6

By the Pythagorean theorem, \(x^2+(x+2)^2=10^2\). Simplifying gives \(2x^2+4x-96=0\), or \(x^2+2x-48=0\). Thus, \((x+8)(x-6)=0\), so \(x=-8\) or \(x=6\). Since a side length must be positive, the correct value is \(x=6\). In length-based problems, always reject a negative root.

Correct answer: C. 7

Let the number be \(x\). Then \((x+3)^2=100\). Taking the positive square root gives \(x+3=10\), so \(x=7\). The negative square root would give \(x=-13\), but the question specifically asks for the positive square root. Exam tip: When solving a squared equation, consider both roots and then apply the condition given in the question.

Correct answer: C. 13

Let the original number be \(x\). Then \((x-4)^2=81\), so \(x-4=9\) or \(x-4=-9\). Hence, \(x=13\) or \(x=-5\). Since the original number is positive, the correct answer is 13. Exam tip: always check both possible signs when solving a squared equation.

Correct answer: C. (8)

The equation is (x^2-6x=16), or (x^2-6x-16=0), giving (x=8). In difference questions, read the order carefully.

Correct answer: B. 9

Translate the wording carefully: the square of the number is 18 more than seven times the number, so x² = 7x + 18. Moving all terms to one side gives x² − 7x − 18 = 0. This factors as (x − 9)(x + 2) = 0, producing x = 9 or x = −2. The question specifically asks for the positive number, so x = 9 and option B is correct. Verification is direct: 9² = 81, while 7 × 9 + 18 = 63 + 18 = 81. The negative root −2 also satisfies the equation algebraically, since (−2)² = 4 and 7(−2) + 18 = 4, but it must be rejected because the required answer is positive. The phrase “18 more than” means add 18, not subtract it.

Correct answer: B. 6

Let the number be x. The phrase “the sum of a number and its square” translates directly to x + x², and the condition says x² + x = 42. Rearranging gives x² + x − 42 = 0. Factoring, we obtain (x + 7)(x − 6) = 0, so the possible values are x = −7 and x = 6. Since the question asks for the positive number, the required answer is 6, which is option B. A quick check confirms that 6 + 6² = 6 + 36 = 42. The negative value −7 also satisfies the equation because −7 + 49 = 42, but it is excluded by the word positive. The other options do not satisfy the condition: 5 + 25 = 30, 7 + 49 = 56, and 8 + 64 = 72.

Correct answer: C. (10)

(x^2+x=110) gives (x^2+x-110=0), whose positive solution is (10). While factoring, check factors of (110).