Word Problems and Applications

If the numbers are (x) and (21-x), then (x(21-x)=110) gives (10) and (11). The smaller number is (10).

Let one number be x; then the other is 25 − x. Thus, x(25 − x) = 156, giving x² − 25x + 156 = 0. Factoring it as
(x − 12)(x − 13) = 0
gives the two numbers 12 and 13. Therefore, the larger number is 13. Exam tip: When the sum and product are given, form the corresponding quadratic equation and find its roots.

Let the smaller number be \(x\). The other number is \(x+3\), so \(x(x+3)=180\), or \(x^2+3x-180=0\). Factoring gives \((x-12)(x+15)=0\), so \(x=12\) or \(x=-15\). Since the number is positive, \(x=12\) is the valid answer. Check: \(12\times15=180\). Exam tip: In such word problems, represent the smaller number by \(x\) and express the other using the given difference, here \(x+3\).

(x(x+5)=204) gives (x=12), so the larger number is (17). While answering, write only the value asked.

Total marks are (x \times x=x^2), so (x^2=121) and (x=11). In context, the number of questions is positive.

The equation is (x(x+2)=120), giving (x=10). For total quantity, multiply number of groups by number in each group.

(x(x+4)=96) gives (x=8). In arrangement questions, total (=) rows (\times) students per row.

The total number of students gives \(x(x-1)=72\), or \(x^2-x-72=0\). Factoring, \((x-9)(x+8)=0\), so \(x=9\) or \(x=-8\). Since the number of benches cannot be negative, \(x=9\) is the valid answer. In word problems, remember to reject roots that have no practical meaning.

The total number of plants gives \(x(x+5)=176\), or \(x^2+5x-176=0\). Factoring, \((x-11)(x+16)=0\), so \(x=11\) or \(x=-16\). Since the number of rows cannot be negative, the answer is 11. Exam tip: In such application problems, retain only the positive root. Option 16 is incorrect because it uses the magnitude of the negative root \(-16\).

(x(x+3)=154) gives (x=11). In a word problem, clearly writing what the variable means is the first step.

If the smaller integer is (x), then (x(x+1)=182), which gives (x=13). Writing consecutive integers as (x) and (x+1) is the correct method.

Let the numbers be (x) and (x+1), then (x(x+1)=210) gives (x=14). So the larger integer is (15).

Taking breadth as (x), (x(x+6)=135), giving (x=9). For a rectangle, take the product of length and breadth for area.

If breadth is (x), then (x(x+11)=180), giving (x=9) and length (20). Write only the dimension asked in the question.

If the side of the square is \(x\) metres, its area is \(x^2\). Thus, \(x^2=196\), giving \(x=\pm14\). Since a length cannot be negative, \(x=14\) metres. Exam tip: when the area of a square is given, find its side by taking the positive square root.

From (x^2=289), the side is (17) m and perimeter is (4x=68) m. First find the side and then apply perimeter.

Let the smaller number be \(x\). Then the larger number is \(x+8\), so \(x(x+8)=240\). This gives \(x^2+8x-240=0\), or \((x-12)(x+20)=0\). Thus, \(x=12\) or \(x=-20\); since the numbers are positive, \(x=12\) is the valid answer. Check: \(12\times20=240\). Remember that 20 is the larger number, not the smaller one.

(x(x+9)=340) gives (x=11), so the larger number is (20). Check the product of both numbers at the end.

(x^2=3x+70) gives (x^2-3x-70=0), and the positive solution is (10). If a positive number is asked, do not take the negative root.

(x^2=2x+120) gives (x^2-2x-120=0) and (x=12). Convert the sentence into an equation in the correct order.