Word Problems and Applications

Correct answer: C. 25

Let the original number be \(x\). Then \((x-11)^2=196\), so \(x-11=\pm14\). This gives \(x=25\) or \(x=-3\). Since the original number is required to be positive, \(x=25\) is correct. In exams, remember to consider both signs when taking the square root and then apply the given condition.

Correct answer: B. 36

For a right triangle, the Pythagorean theorem states that the sum of the squares of the two perpendicular legs equals the square of the hypotenuse. Therefore x² + (x + 12)² = 60². Expanding gives 2x² + 24x + 144 = 3600, or x² + 12x − 1728 = 0. Factoring gives (x − 36)(x + 48) = 0, so x = 36 or −48. Lengths must be positive, hence x = 36 and option B is correct. Checking: 36² + 48² = 1296 + 2304 = 3600 = 60². The other choices do not satisfy the Pythagorean equation. The governing concept is using a geometric relation to form and solve a quadratic equation, while retaining only a physically meaningful positive length.

Correct answer: B. 27 मीटर

By the Pythagorean theorem, \(x^2+(x+9)^2=45^2\). Simplifying gives \(x^2+9x-972=0\), which factors as \((x-27)(x+36)=0\). Thus, \(x=27\) or \(x=-36\). Since a length cannot be negative, the shorter leg is 27 m. Exam tip: Reject any negative root when the variable represents a length.

Correct answer: A. (15) m

If breadth is (x) and length is (x+7), then (2(x+x+7)=74), giving (x=15). In perimeter questions, use (2(l+b)).

Correct answer: C. (27) m

Direct answer: C, 27 m. Let breadth be x m. Then length is x+9. Perimeter is 2(length+breadth), so 2[(x+9)+x]=90. Dividing by 2 gives 2x+9=45, hence 2x=36 and x=18. Therefore length=18+9=27 m. Option A, 18 m, is the breadth, not the requested length. Option B, 22 m, does not differ from a matching breadth by 9 while giving the stated perimeter. Option C, 27 m, satisfies the conditions: 27-18=9 and 2(27+18)=90. Option D, 30 m, would require breadth 21 m, giving perimeter 102 m. Memory cue: first divide the perimeter by 2 to get length+breadth, then use the difference.

Correct answer: B. (17)

The equation is (x(x+8)=425), giving (x=17). For total number, multiply rows and number per row.

Correct answer: A. 17

The total number of books equals the number of racks multiplied by the books in each rack, so \(x(x+6)=391\). Thus, \(x^2+6x-391=0\), which factors as \((x-17)(x+23)=0\). The roots are \(x=17\) and \(x=-23\); since the number of racks cannot be negative, \(x=17\) is correct. Exam tip: In real-life word problems, reject a negative root when the variable represents a quantity.

Correct answer: B. 20

The total-seat condition gives x(x + 5) = 500. Thus, x² + 5x − 500 = 0 = (x − 20)(x + 25), so x = 20 or x = −25. Since the number of rows cannot be negative, the valid answer is 20 rows. In such word problems, reject roots that are not meaningful in the given context.

Correct answer: C. 26

The total number of plants gives \(x(x+11)=702\). Thus, \(x^2+11x-702=0\), which factors as \((x-26)(x+27)=0\). So \(x=26\) or \(x=-27\); a number of rows cannot be negative, so \(x=26\) is correct. Remember that 27 is the number of plants in each row, not the number of rows.

Correct answer: B. (20)

If there are \(x\u0005 rows and each row contains \(x+12\u0005 chairs, the total number of chairs is their product. Thus \(x(x+12)=640\u0005. Expanding gives \(x^2+12x-640=0\u0005. This factors as \((x-20)(x+32)=0\u0005, so the algebraic values are \(x=20\u0005 and \(x=-32\u0005.

A number of rows cannot be negative, so only \(x=20\u0005 is meaningful. There are then 32 chairs per row, and \(20\times32=640\u0005 confirms the result. Therefore option B is correct. The negative root is rejected because it has no physical meaning in this counting situation.

Correct answer: A. 17

The total number of questions gives the equation x(x + 7) = 408, or x² + 7x − 408 = 0. Factoring gives (x − 17)(x + 24) = 0, so x = 17 or x = −24. Since the number of days cannot be negative, x = 17 is the valid answer. As an exam tip, verify it by checking that 17 × 24 = 408.

Correct answer: B. (20)

The equation is (x(x+9)=700), giving (x=20). Multiply the number of groups by objects per group.

Correct answer: C. 21

Let the number be \(x\). Then \(x^2+71=512\), so \(x^2=512-71=441=21^2\). Hence \(x=\pm21\), and the positive number is \(21\). Option 20 is incorrect because \(20^2+71=471\), not 512. Exam tip: Taking a square root gives both positive and negative roots, but choose the positive root when the question asks for a positive number.

Correct answer: C. 24

Let the number be \(x\). Then \(x^2-119=457\), so \(x^2=576=24^2\). Thus \(x=\pm24\), but the positive number is \(24\). In the exam, use the word “positive” to reject the negative root.

Correct answer: C. (27) cm

The correct answer is option C: 27 cm. Equal areas mean the square’s area equals the rectangle’s area. The square side is 18 cm, so its area is 18^2=324 cm². Let the rectangle’s length be l cm. Its breadth is 12 cm, so its area is 12l. Equate the areas: 12l=324. Dividing by 12 gives l=27 cm. Option A, 18 cm, would give rectangle area 12×18=216 cm², not 324. Option B, 24 cm, gives 288 cm², so it is too short. Option C is correct because 12×27=324 cm², exactly equal to the square’s area. Option D, 36 cm, gives 432 cm², so it is too long. Notice that the square’s perimeter is irrelevant; only its area is needed. Also keep units consistent: multiplying centimetres by centimetres gives square centimetres. Memory cue: equal areas → calculate both area formulas → equate them → divide by the known rectangle side.

Correct answer: C. 27

The area of a square is side × side, so x² = 729. Since 729 = 27², the possible roots are 27 and −27; a side length must be positive, so x = 27 metres.

Correct answer: B. 16 cm

Let the breadth be \(x\) cm. Then the length is \(x+4\) cm, so \(x(x+4)=320\), or \(x^2+4x-320=0\). Factoring gives \((x-16)(x+20)=0\), so \(x=16\) or \(x=-20\). Since a dimension cannot be negative, the breadth is 16 cm. Exam tip: In measurement problems, reject any negative root.

Correct answer: C. 23 cm

Let the breadth be \(x\) cm. Then the length is \(x+6\) cm, so \(x(x+6)=667\), giving \(x^2+6x-667=0\). Factoring, \((x-23)(x+29)=0\), gives \(x=23\) or \(x=-29\). A length or breadth cannot be negative, so the breadth is 23 cm. Option D is the length, not the breadth. Exam tip: verify the answer by multiplying the breadth and the corresponding length to check the given area.

Correct answer: B. 19

Let the smaller number be x. The number that is 15 more than it is x + 15. Their sum is 53, so x + (x + 15) = 53. Simplifying gives 2x + 15 = 53, then 2x = 38 and x = 19. Therefore option B is correct. A direct check gives 19 + 34 = 53, and 34 is exactly 15 more than 19. Option D is the larger number rather than the smaller one. Options A and C produce sums of 49 and 57, respectively, when paired with the number 15 greater than them. This problem illustrates that a word problem may lead to a linear equation rather than a quadratic one; the topic is still an application of algebraic equations, but no square term is needed.

Correct answer: B. A number and (3) more than it have product (88)

Option (B) forms (x(x+3)=88), which is quadratic. When a variable is multiplied by a variable expression, an (x^2) term appears.

Correct answer: B. (x(x+10)=651)

Area of a rectangle is length (\times) breadth, so (x(x+10)=651) is correct. Identifying the formula is the first step.

Correct answer: B. 25

The total number of plants gives \(x(x-4)=525\), or \(x^2-4x-525=0\). Factoring, \((x-25)(x+21)=0\), so \(x=25\) or \(x=-21\). Since the number of rows cannot be negative, \(x=25\) is the valid answer. In such word problems, always reject a negative value for a physical quantity.

Correct answer: B. (21)

The equation is (x(x+2)=483), giving (x=21). For total count, multiply students and questions per student.

Correct answer: A. 35 years

Let the father's age be x. Then x(x − 30) = 175, so x² − 30x − 175 = 0 = (x − 35)(x + 5). The roots are 35 and −5. An age cannot be negative, so the father's age is 35 years; the son's age is 5 years, and their product is 175.

Correct answer: B. 15

Let the original number be \(x\). Then \((x+2)(x+3)=306\), which gives \(x^2+5x-300=0\). Factoring, \((x+20)(x-15)=0\), so \(x=-20\) or \(x=15\). Since the question specifies a positive number, the original number is \(15\). Exam tip: Form the algebraic equation first and then select the root that satisfies the condition in the question.