Direct And Inverse Proportion! Math Trivia Questions Quiz

Explanation
The given statement "a is proportional to the cube of j" means that the value of a is directly related to the cube of j. In other words, if j is multiplied by a certain factor, then a will be multiplied by the cube of that factor.

In the first equation, we are given that when j = 8, a = 4096. Since a is proportional to the cube of j, we can write the equation as a = k * j^3, where k is the constant of proportionality. Plugging in the values, we get 4096 = k * 8^3. Solving for k, we find that k = 8. Therefore, the first equation can be completed as a = 8 * j^3.

In the second equation, we are asked to find the value of a when j = 1. Using the completed equation from the first part, we can substitute j = 1 and find that a = 8 * 1^3 = 8.

Explanation
In an inverse variation, the product of the two variables remains constant. To complete the equation, we can write it as f * r = k, where k is the constant. Given that f = 4 when r = 15, we can substitute these values into the equation to find k: 4 * 15 = k, which gives us k = 60. Therefore, the equation is f * r = 60. To find the value of r when f = 60, we can rearrange the equation to solve for r: r = k / f = 60 / 60 = 1. Therefore, when f = 60, r = 1.

Explanation
The equation for the direct variation between h and a is h = ka^2, where k is the constant of variation. Since h = 64 when a = 2, we can substitute these values into the equation to find the value of k. 64 = k(2^2) => 64 = 4k => k = 16. Therefore, the equation is h = 16a^2. To find h when a = 5, we substitute this value into the equation: h = 16(5^2) = 16(25) = 400.

Explanation
The equation for the inverse proportion is g * y = k, where k is a constant. To complete the equation, we can substitute the given values g = 6 and y = 4 into the equation: 6 * 4 = k, which gives us k = 24. Therefore, the complete equation is g * y = 24. To find the value of y when g = 8, we can rearrange the equation to solve for y: y = k / g. Substituting the values k = 24 and g = 8 into the equation, we get y = 24 / 8 = 3. Hence, when g = 8, y = 3.

Explanation
When completing the equation t = [Blank], the missing value is 9. When t = 72, the corresponding value of q is 8.

Explanation
The equation for the inverse variation relationship between p and u can be written as p = k/u^2, where k is a constant. To find the value of k, we can substitute the given values of p and u into the equation. When p = 8 and u = 14, we get 8 = k/14^2. Solving for k, we find k = 1568. To find the value of p when u = 2, we can use the equation p = 1568/2^2, which simplifies to p = 392.

Explanation
We can solve this problem by using the concept of direct proportionality. The equation m = kb^3 represents a direct proportionality between m and b^3, where k is a constant. We are given that m = 24 when b = 2, so we can substitute these values into the equation to find the value of k. 24 = k(2^3) = 8k. Solving for k, we find that k = 3. Now, we can use this value of k to find the value of b when m = 2187. 2187 = 3(b^3). Dividing both sides by 3, we get b^3 = 729. Taking the cube root of both sides, we find that b = 9. Therefore, the answer is 27.

Explanation
In this question, we are given that c is inversely proportional to v. This means that as v increases, c decreases, and vice versa. We are also given that when v = 18, c = 5. To find the value of c when v = 4, we can set up a proportion using the inverse relationship. Since c and v are inversely proportional, we can write the proportion as c1v1 = c2v2, where c1 = 5, v1 = 18, c2 is the value we need to find, and v2 = 4. Solving for c2, we get c2 = (c1v1) / v2 = (5 * 18) / 4 = 22.5. Therefore, when v = 4, c = 22.5.

Explanation
Since w is directly proportional to n, we can use the concept of direct variation to find the value of w when n is 6. We can set up a proportion using the given information: 200/10 = w/6. Cross-multiplying, we get 10w = 1200, and dividing both sides by 10, we find that w = 120. Therefore, when n is 6, w is equal to 120.

Explanation
The given problem states that s varies inversely as z. This means that as z increases, s decreases, and vice versa. The equation s = 7 when z = 12 establishes a relationship between s and z. To find the value of z when s = 28, we can set up a proportion using the given equation: s/z = 7/12. Solving for z, we find that z = 12(7/28) = 3. Therefore, when s = 28, z = 3.

Explanation
The given question states that e is proportional to the square of g. This means that as g increases or decreases, e will increase or decrease by the square of that change. The equation e = 405 when g = 9 tells us that when g is 9, e is 405. Therefore, we can find e when g is 5 by using the proportionality relationship. Since 5 is 4 units less than 9, we square that change (4^2) and apply it to e. 405 - (4^2) = 405 - 16 = 389. Therefore, when g is 5, e is 389.

Explanation
In this question, we are given that j varies inversely as h. This means that as j increases, h decreases, and vice versa. We are also given that when h = 2, j = 6. To find the value of h when j = 1.5, we can set up an inverse variation equation: j1 * h1 = j2 * h2, where j1 = 6, h1 = 2, j2 = 1.5, and h2 is what we need to find. Plugging in the values, we get 6 * 2 = 1.5 * h2. Solving for h2, we find that h2 = 8. Therefore, when j = 1.5, h = 8.

Explanation
When r = 6, y = 1. Therefore, the relationship between r and y can be represented by the equation y = 1/3r. When r = 2, substituting the value into the equation gives y = 1/3(2) = 2/3 = 0.67. However, the given answer is 9, which does not match the equation or the given information. Therefore, the correct answer is not 9.

Explanation
The relationship between u and c is given as u is proportional to c cubed. This means that u is equal to some constant multiplied by c cubed. We are given that u is equal to 640 when c is equal to 4. By substituting these values into the equation, we can solve for the constant. Once we have the constant, we can use it to find the value of c when u is equal to 3430. The value of c is 7.

Explanation
The given statement "p varies directly as t^2" indicates that p and t^2 are directly proportional. This means that as t^2 increases, p also increases, and vice versa. Using the given information, we can set up a proportion to find the value of t when p = 63. Using the formula for direct variation, we have (p1/t1^2) = (p2/t2^2), where p1 = 700, t1 = 10, and p2 = 63. Solving for t2, we get t2^2 = (10^2 * 63)/700. Simplifying this equation gives us t2^2 = 9, which means t2 can be either 3 or -3.

Explanation
The given information states that "a varies inversely as the square of x." This means that as x decreases, a will increase. We are given that when x = 18, a = 5. Therefore, when x decreases to 6, a will increase. Since we know that a = 5 when x = 18, we can conclude that when x = 6, a will still be 5. Therefore, the answer is 5.

Explanation
The value of b is directly proportional to the value of w. When w is 4, b is 16. So, we can find the value of b when w is 8 by using the formula for direct proportionality. We can set up a proportion: 4/w = 16/8. Solving for w, we find that w = 2.

Explanation
Since n is inversely proportional to f, it means that as f decreases, n increases. We can use the formula for inverse proportionality, n = k/f, where k is a constant. We can solve for k by substituting the given values of n and f into the equation: 21 = k/12. Solving for k, we get k = 252. Now we can use the value of k to find n when f = 9: n = 252/9 = 28. Therefore, when f = 9, n = 28.

Explanation
When v = 36 and s = 16, we can use the equation v = s^2 to find the relationship between v and s. Plugging in the given values, we get 36 = 16^2. Simplifying this equation, we find that 36 = 256. Therefore, when v = 9, s must be equal to 256 for the equation v = s^2 to hold true.

Explanation
When m = 6144 and z = 8, we can see that there is a linear relationship between m and z. To find the value of z when m = 12000, we can set up a proportion using the given values: 6144/8 = 12000/z. Cross-multiplying and solving for z, we get z = 10. Therefore, when m = 12000, z = 10.