answer:Let's solve the equation step by step: 60% of (x - y) = 30% of (x + y) First, convert the percentages to decimals: 0.60(x - y) = 0.30(x + y) Now, distribute the decimals into the parentheses: 0.60x - 0.60y = 0.30x + 0.30y Next, let's move all terms involving x to one side and all terms involving y to the other side: 0.60x - 0.30x = 0.60y + 0.30y Combine like terms: 0.30x = 0.90y Now, divide both sides by 0.30 to solve for x in terms of y: x = 0.90y / 0.30 x = 3y Now we have x in terms of y. To find what percent of x is y, we can set up the following ratio: y / x = y / (3y) When we simplify this, we get: y / (3y) = 1/3 Convert the fraction to a percentage: 1/3 = 0.3333... As a percentage, this is approximately 33.33%. Therefore, y is approximately boxed{33.33%} of x.

answer:Let's denote the sequence as (5, a, b, c, d, e, f, 8). Step-by-Step Solution: 1. **Sum of First Three Numbers:** [ 5 + a + b = 20 ] This gives us: [ a + b = 15 quad text{(1)} ] 2. **Sum of the Second, Third, and Fourth Numbers:** [ a + b + c = 20 ] Substitute (a + b) from equation (1): [ 15 + c = 20 implies c = 5 ] 3. **Sum of the Third, Fourth, and Fifth Numbers:** [ b + c + d = 20 ] Substitute (c = 5): [ b + 5 + d = 20 implies b + d = 15 quad text{(2)} ] 4. **Sum of the Fourth, Fifth, and Sixth Numbers:** [ c + d + e = 20 ] Substitute (c = 5) and use equation (2): [ 5 + d + e = 20 implies d + e = 15 quad text{(3)} ] 5. **Sum of the Fifth, Sixth, and Seventh Numbers:** [ d + e + f = 20 ] From (3), substitute (d + e): [ 15 + f = 20 implies f = 5 ] 6. **Sum of the Sixth, Seventh, and Eighth Numbers:** [ e + f + 8 = 20 ] Substitute (f = 5): [ e + 5 + 8 = 20 implies e = 7 ] 7. **Finding (a) and (b):** From equation (3): [ d + e = 15 implies 5 + 7 = 15 implies 7 = d ] From equation (2): [ b + d = 15 implies b + 7 = 15 implies b = 7 ] From equation (1): [ a + b = 15 implies a + 7 = 15 implies a = 8 ] Finally, we get the sequence: [ 5, 8, 7, 5, 8, 7, 5, 8 ] Conclusion: [ boxed{5,8,7,5,8,7,5,8} ]

answer:For a quadratic equation ax^2 + bx + c = 0 to have exactly one solution, the discriminant must be zero. The discriminant is given by b^2 - 4ac. 1. For the equation 3x^2 - 7x + m = 0, the discriminant is (-7)^2 - 4 cdot 3 cdot m = 49 - 12m. 2. Setting the discriminant to zero gives: [ 49 - 12m = 0 ] 3. Solving for m: [ 12m = 49 quad Longrightarrow quad m = frac{49}{12} ] Thus, m = boxed{frac{49}{12}}.

answer:To find all positive integer solutions to the equation (3m + 4n = 5k), we can proceed as follows: 1. **Analyzing the equation modulo 3:** [ 5k equiv (-1)^k pmod{3} ] because (5 equiv -1 pmod{3}). Since (4n equiv 1 pmod{3}) (because (4 equiv 1 pmod{3})), we get: Substituting into the original equation: [ 4n equiv 5k - 3m equiv (-1)^k pmod{3} ] Therefore, (5k equiv -1 pmod{3}). For this to hold, (k) must be even. Put (k = 2K). 2. **Analyzing the equation modulo 4:** [ 3m equiv (-1)^m pmod{4} ] because (3 equiv -1 pmod{4}). Since (5k equiv 1 pmod{4}) (because (5^2 = 25 equiv 1 pmod{4})), we get: Substituting into the original equation: [ 5k equiv 3m + 4n equiv 1 pmod{4} ] Therefore, (3mequiv 1 pmod{4}). For this to hold, (m) must be even. Put (m = 2M). 3. **Substitution and factorization:** Substituting (m = 2M) and (k = 2K) into the equation, we get: [ 3(2M) + 4n = 5(2K) ] which simplifies to: [ 6M + 4n = 10K ] Dividing the entire equation by 2: [ 3M + 2n = 5K ] 4. **Further factorization:** Looking deeper: [ 5^k + 3^m = 2^a + 2^b ] 5. **Considering powers and verifying the equation:** Separate terms and recognize that: [ 5K - 3M = 2^b ] [ 5K + 3M = 2^(a+b) ] Subtracting these two, we get: [ 2 cdot 3M = 2^b (2^a - 1)Rightarrow 3M = 2^{b-1}(2^a - 1) ] Here, (b) must be 1. Then the equation simplifies to: [ 3M = 2^a - 1 ] Conclusion, matching mod 8, (3M equiv 3Mrightarrow Then solving integers for all a and M: Therefore (a = 2, and M = 1). 6. **Conclusion:** To have 3M = 2 + 2n, validating the results: So we have (M = 1), (n = 2) [ 3(2) + 4(2) = 5k 6 + 8 = 20 = 5 cdot 2 Hence k must be 2. Thus, the solution satisfying all conditions of the equation is: [ m =1, n =2, k =2 boxed{text{m = 1, n = 2, k = 2}}