Word Problems and Applications

Let the smaller number be (x) and the larger be (x+6), then (x(x+6)=160) gives (10) and (16). The larger number is (16).

Put the smaller number as (x) and the larger as (x+8), then (x(x+8)=425) gives (x=17). Add the difference in the correct form.

Let the breadth be \(x\) cm. Then the length is \(x+3\) cm. Using the area formula, \(x(x+3)=154\), so \(x^2+3x-154=0\). Factoring gives \((x-11)(x+14)=0\), yielding \(x=11\) or \(x=-14\). Since length and breadth must be positive, \(-14\) is not physically valid. Therefore, the breadth is 11 cm. Exam tip: In dimension-based word problems, reject any negative root.

If breadth is (x), then (x(x+5)=126), so (x=9) and length (14). Before answering, check the word asked.

If the side of the square is \(x\) cm, its area is \(x^2\). Thus, \(x^2=361\), so \(x=\sqrt{361}=19\). A side length cannot be negative, so \(-19\) is rejected. Exam tip: When the area of a square is given, find its side by taking the square root.

(x^2=400) gives side (20) m and perimeter (4x=80) m. Area and perimeter formulas are different.

The equation is (x(x+6)=187), giving (x=11). For total number, multiply rows and number per row.

(x(x+1)=272) gives (x=16). In an arrangement, total (=) number of groups (\times) objects per group.

The total-seat condition gives \(x(x-2)=224\). Thus, \(x^2-2x-224=0\), which factors as \((x-16)(x+14)=0\). Hence, \(x=16\) or \(x=-14\). Since the number of rows cannot be negative, the valid answer is 16. Exam tip: In word problems, always check the roots against the conditions of the real situation, such as being a positive integer.

The total number of flowers gives \(x(x+7)=330\), or \(x^2+7x-330=0\). Factoring gives \((x-15)(x+22)=0\), so \(x=15\) or \(x=-22\). Since the number of rows cannot be negative, the valid answer is 15. In word problems, always reject a root that is impossible in the given context.

The total number of chairs gives \(x(x+10)=416\), or \(x^2+10x-416=0\). Factoring, \((x-16)(x+26)=0\), so \(x=16\) or \(x=-26\). Since the number of rows cannot be negative, the valid answer is 16. In such word problems, always check whether the root is meaningful in context.

The total number of trees gives the equation \(x(x+4)=221\). Thus, \(x^2+4x-221=0\), which factors as \((x-13)(x+17)=0\). Hence, \(x=13\) or \(x=-17\), but the number of rows cannot be negative, so the correct answer is 13. Exam tip: In word problems, check the roots against the given context and reject impossible negative values.

Total questions = number of days × questions solved per day. Thus, (x(x+3)=130), or (x^2+3x-130=0). Factoring gives ((x+13)(x-10)=0), so (x=10) or (x=-13). The number of days cannot be negative, so the answer is 10 days. Exam tip: In word problems, reject a negative root when the quantity represents time, length, or number of objects.

The equation is (x(x+5)=216), which gives (x=12). In group-based questions, the total number is formed by multiplication.

(x^2+6=202) gives (x^2=196), so the positive (x=14). While taking square root, check the condition in the question.

(x^2-25=119) gives (x^2=144), so (x=12). In subtraction questions, place the sign carefully.

Let the number be \(x\). Then \(x^2+39=208\), so \(x^2=208-39=169=13^2\). Thus, \(x=13\) or \(-13\); since the question asks for the positive number, the correct answer is 13. Exam tip: isolate the squared term first, then take the square root and check the sign condition.

Taking breadth as (x), (x(x+18)=648), giving (x=18). In a rectangular region, make area the base equation.

(x(x+21)=550) gives breadth (25) and length (46). After solving, identify length and breadth separately.

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