Word Problems and Applications

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.

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

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.

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

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

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

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.

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

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.

(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.

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.

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

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

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.

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.

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.

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

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.

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.

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.

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.

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

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.

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.

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