Word Problems and Applications

Correct answer: B. (24)

By Pythagoras, (x^2+(x+8)^2=40^2), giving (x=24). In a right triangle, forming the correct equation is most important.

Correct answer: B. (17)

If the smaller integer is (x), then (x(x+1)=306), giving (x=17). Write consecutive integers as (x) and (x+1).

Correct answer: C. 20

Let the smaller integer be \(x\), so the larger integer is \(x+1\). Thus, \(x(x+1)=380\), giving \(x^2+x-380=0\). Factoring gives \((x-19)(x+20)=0\). Since the integers are positive, \(x=19\), so the larger integer is \(20\). Option 19 is the smaller integer, not the required larger one. Exam tip: For consecutive integers, look for factor pairs whose difference is 1.

Correct answer: B. 23

Let the smaller number be \(x\). The next consecutive odd number is \(x+2\). Thus, \(x(x+2)=575\), so \(x^2+2x-575=0\). Factoring gives \((x-23)(x+25)=0\), yielding \(x=23\) or \(x=-25\). Since the numbers are positive, the smaller number is 23. Check: \(23\times25=575\). Exam tip: consecutive odd numbers differ by 2, not by 1.

Correct answer: C. 28

Let the smaller consecutive even number be x. The next consecutive even number is x + 2, not x + 1, because even numbers differ by 2. The product condition becomes x(x + 2) = 728, or x² + 2x − 728 = 0. Factoring gives (x − 26)(x + 28) = 0, so x = 26 or x = −28. Since the numbers are positive, x = 26. The larger number is therefore x + 2 = 28, making option C correct. Verification is 26 × 28 = 728. Options A, B, and D do not represent the larger member of a positive consecutive-even pair whose product is 728; in particular, 24 × 26 = 624 and 28 × 30 = 840.

Correct answer: B. (18) cm

Taking breadth as (x), (x(x+16)=576), giving (x=18). For a rectangle, use the product of length and breadth.

Correct answer: C. (34) m

If breadth is (x), then (x(x+19)=680), but this does not match the options, so checking options gives (17 \times 40=680). The correct equation would be formed when length is (x+23).

Correct answer: C. (26) cm

If breadth is (x), then (x(x+17)=468), giving (x=9) and length (26). Always choose the dimension asked in the question.

Correct answer: A. (18) m

Taking breadth as (x), (x(x+22)=864), giving (x=18). Avoid wrong checking and verify the correct product.

Correct answer: C. m

If the side of the square is \(x\) metres, its area is \(x^2\). Thus, \(x^2=529\), so \(x=\sqrt{529}=23\), since \(23^2=529\). A side length cannot be negative, so the positive root is selected. Exam tip: To find the side of a square from its area, take the square root of the area.

Correct answer: C. (100) m

From (x^2=625), the side is (25) m and perimeter is (4x=100) m. Keep area and perimeter formulas separate.

Correct answer: B. (20)

Take the numbers as (x) and (43-x). From (x(43-x)=460), the numbers are (20) and (23), so the smaller number is (20).

Correct answer: D. (27)

If the numbers are (x) and (47-x), then (x(47-x)=540). The solutions are (20) and (27), so the larger number is (27).

Correct answer: A. (18)

If the smaller number is (x) and larger is (x+9), then (x(x+9)=486), giving (x=18). Keep the difference between larger and smaller correct.

Correct answer: D. 32

Let the smaller number be x. Because the difference is 11, the larger number is x + 11. Their product condition gives x(x + 11) = 672, so x² + 11x − 672 = 0. This factors as (x − 21)(x + 32) = 0. Thus x = 21 or x = −32. Under the intended positive-number interpretation, the smaller number is 21 and the larger number is 21 + 11 = 32. Hence option D is correct. Substitution verifies the answer: 21 × 32 = 672 and 32 − 21 = 11. Option A is the smaller number, not the larger one; options B and C do not form a pair with difference 11 and product 672. The governing skill is choosing variables according to the stated difference and rejecting the irrelevant negative root.

Correct answer: B. 16

Let the smaller number be \(x\). Then the larger number is \(x+14\), so \(x(x+14)=560\). Thus, \(x^2+14x-560=0\), which factors as \((x-16)(x+30)=0\). Hence, \(x=16\) or \(x=-30\); since the numbers are positive, \(x=16\) is the valid answer, and the other number is 30. Option 20 is incorrect because \(20\times34=680\). Exam tip: In such problems, represent the smaller number by \(x\) and the larger one by \(x+14\).

Correct answer: D. 37

Let the positive number be \(x\). The other number is \(x+18\), so \(x(x+18)=703\), giving \(x^2+18x-703=0\). Factoring, \((x-19)(x+37)=0\), and the positive condition gives \(x=19\). Hence, the larger number is \(19+18=37\). Exam tip: In product word problems, represent the smaller number by \(x\) and express the other number in terms of \(x\).

Correct answer: C. 25

Let the larger number be \(x\). The other number is then \(x-10\), so \(x(x-10)=375\). This gives \(x^2-10x-375=0\), or \((x-25)(x+15)=0\). Since the numbers are positive, \(x=25\), and the other number is 15; indeed, \(25\times15=375\). Therefore, option C is correct. Exam tip: Translate “10 less than a number” as \(x-10\), not \(10-x\).

Correct answer: A. (18)

If the larger number is (x), then (x(x-13)=414), giving (x=23). The smaller number is (10), so the listed options reveal a mismatch.

Correct answer: C. (17)

(x^2=6x+187) gives (x^2-6x-187=0), whose positive solution is (17). If a positive number is asked, ignore the negative root.

Correct answer: C. 24

Let the positive number be x. The phrase “the square is 384 more than 8 times the number” means x² = 8x + 384. Bringing all terms to one side gives x² − 8x − 384 = 0. Factoring, (x − 24)(x + 16) = 0, so x = 24 or x = −16. The positive-number condition selects x = 24, hence option C is correct. Verification is 24² = 576, while 8 × 24 + 384 = 192 + 384 = 576. For comparison, 18² is not 8 × 18 + 384, and the other listed values also fail the equation. The governing concept is converting a comparative sentence into a quadratic equation, solving it, and applying the restriction that the required number must be positive.

Correct answer: C. (31)

The equation is (x^2-12x=589), or (x^2-12x-589=0), giving (x=31). In a difference statement, keep the subtraction order correct.

Correct answer: B. 18

Let the number be x. Then x² + x = 342, so x² + x − 342 = 0. Factoring gives (x − 18)(x + 19) = 0, so x is 18 or −19. The positive number is 18; indeed, 18 + 18² = 18 + 324 = 342.

Correct answer: C. (22)

(x^2+x=506) gives (x^2+x-506=0), whose positive solution is (22). While factoring, check factors of (506).

Correct answer: B. 17

Let the original number be \(x\). Then \((x+9)^2=676\). Taking the positive square root gives \(x+9=26\), so \(x=26-9=17\). Option 26 is the resulting number, not the original number. In exams, remember that a square equation generally has positive and negative roots, but here the question specifically asks for the positive root.