Question

question 16
\\(\frac{5}{50} \div \frac{15}{30}\\)

Explanation:

Step1: Recall fraction division rule

To divide fractions, multiply by the reciprocal: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\). So, \(\frac{5}{50} \div \frac{15}{30} = \frac{5}{50} \times \frac{30}{15}\).

Step2: Simplify before multiplying

Simplify numerators and denominators: \(5\) and \(15\) have a GCD of \(5\), so \(\frac{5\div5}{15\div5}=\frac{1}{3}\); \(30\) and \(50\) have a GCD of \(10\), so \(\frac{30\div10}{50\div10}=\frac{3}{5}\). Now the expression is \(\frac{1}{10} \times \frac{3}{3}\) (wait, better to simplify step by step: \(\frac{5}{50}=\frac{1}{10}\), \(\frac{30}{15}=2\). So \(\frac{1}{10} \times 2=\frac{2}{10}=\frac{1}{5}\)? Wait, no, let's do it correctly: \(\frac{5}{50} \times \frac{30}{15}\). \(5\) and \(15\): \(5\) goes into \(5\) once, \(15\) three times. \(30\) and \(50\): \(10\) goes into \(30\) three times, \(50\) five times. So now we have \(\frac{1}{5} \times \frac{3}{3}\)? No, wait, \(\frac{5}{50} = \frac{1}{10}\), \(\frac{30}{15}=2\). So \(\frac{1}{10} \times 2 = \frac{2}{10} = \frac{1}{5}\)? Wait, no, let's multiply numerators: \(5\times30 = 150\), denominators: \(50\times15 = 750\). Then \(\frac{150}{750}\) simplifies by dividing numerator and denominator by \(150\): \(\frac{150\div150}{750\div150}=\frac{1}{5}\)? Wait, no, \(150\div750 = \frac{1}{5}\)? Wait, \(150\times5=750\), yes. Wait, but let's check again: \(\frac{5}{50} \div \frac{15}{30} = \frac{5}{50} \times \frac{30}{15}\). \(5\) and \(15\): divide by \(5\), get \(1\) and \(3\). \(30\) and \(50\): divide by \(10\), get \(3\) and \(5\). So now it's \(\frac{1}{5} \times \frac{3}{3}\)? No, \(\frac{5\div5}{50} \times \frac{30}{15\div5}\) is \(\frac{1}{50} \times \frac{30}{3}\). Then \(30\div3 = 10\), so \(\frac{1}{50} \times 10 = \frac{10}{50} = \frac{1}{5}\). Wait, but let's do it with GCD: \(\frac{5\times30}{50\times15}=\frac{150}{750}\). GCD of \(150\) and \(750\) is \(150\), so \(\frac{150\div150}{750\div150}=\frac{1}{5}\). Wait, but maybe I made a mistake. Wait, \(\frac{5}{50} = \frac{1}{10}\), \(\frac{15}{30} = \frac{1}{2}\). So dividing \(\frac{1}{10} \div \frac{1}{2} = \frac{1}{10} \times 2 = \frac{2}{10} = \frac{1}{5}\). Yes, that's correct. Wait, but the options: let's check the options. Wait, the options include \(1/5\)? Wait, the options given: 3: 5/8, 4:15/28, 5:1/7, 6:8/15, 7:6/25, 8:2/7, 9:2/5, 10:1/4, 11:9/25, 12:1, 13:3/4, 14:3/8, 15:1/3, 16: (blank), 17: (blank), 18: (blank). Wait, maybe I miscalculated. Wait, let's do it again: \(\frac{5}{50} \div \frac{15}{30}\). Reciprocal of \(\frac{15}{30}\) is \(\frac{30}{15}\). Multiply: \(\frac{5\times30}{50\times15} = \frac{150}{750}\). Simplify by dividing numerator and denominator by 150: \(150\div150=1\), \(750\div150=5\). So \(\frac{1}{5}\). But \(\frac{1}{5}\) is not in the options? Wait, no, option 9 is 2/5, 10 is 1/4, 15 is 1/3. Wait, maybe I made a mistake. Wait, \(\frac{5}{50} = \frac{1}{10}\), \(\frac{15}{30} = \frac{1}{2}\). Dividing \(\frac{1}{10} \div \frac{1}{2} = \frac{1}{10} \times 2 = \frac{2}{10} = \frac{1}{5}\). But \(\frac{1}{5}\) is not listed? Wait, maybe the problem is \(\frac{5}{50} \div \frac{15}{30}\) – wait, 5/50 is 1/10, 15/30 is 1/2. 1/10 divided by 1/2 is 1/5. But the options: let's check again. The options are 3:5/8, 4:15/28, 5:1/7, 6:8/15, 7:6/25, 8:2/7, 9:2/5, 10:1/4, 11:9/25, 12:1, 13:3/4, 14:3/8, 15:1/3, 16: (blank), 17: (blank), 18: (blank). Wait, maybe I misread the problem. Is it 5/50 divided by 15/30? Wait, 5/50 is 1/10, 15/30 is 1/2. 1/10 divided by 1/2 is 1/5. But 1/5 is not an option? Wait, maybe the problem is 5/50 divided by 15/30, but maybe I made a mistake in reciprocal. Wait, no, division of fractions is multiply by reciprocal. So \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\). So that's correct. Wait, maybe the problem is 5/50 divided by 15/30, but 5/50 is 1/10, 15/30 is 1/2. 1/10 * 2/1 = 2/10 = 1/5. Hmm. Maybe the options are miswritten, or I misread the numbers. Wait, the original problem: 5/50 ÷ 15/30. Let's compute numerator: 5*30=150, denominator:50*15=750. 150/750 = 1/5. So maybe the answer is 1/5, but it's not in the options? Wait, maybe the problem is 5/50 ÷ 15/30, but 5/50 is 1/10, 15/30 is 1/2. 1/10 divided by 1/2 is 1/5. Alternatively, maybe the problem is 5/50 ÷ 15/30, but 5 and 15: GCD 5, 30 and 50: GCD 10. So \(\frac{5\div5}{15\div5} = \frac{1}{3}\), \(\frac{30\div10}{50\div10} = \frac{3}{5}\). Then \(\frac{1}{10} \times \frac{3}{3}\)? No, that's not right. Wait, maybe I should do it as \(\frac{5}{50} \times \frac{30}{15} = \frac{5\times30}{50\times15} = \frac{150}{750} = \frac{1}{5}\). So the answer is \(\frac{1}{5}\), but since it's not in the options, maybe I made a mistake. Wait, maybe the problem is 5/50 ÷ 15/30, but 5/50 is 1/10, 15/30 is 1/2. 1/10 divided by 1/2 is 1/5. Alternatively, maybe the problem is 5/50 ÷ 15/30, but 5/50 is 1/10, 15/30 is 1/2. 1/10 * 2 = 1/5. So perhaps the options are missing, but according to the calculation, the answer is \(\frac{1}{5}\). Wait, but maybe I misread the fractions. Is it 5/50 or 5/5? No, the problem says 5/50. Hmm.

Answer:

\(\frac{1}{5}\) (Note: If this is not in the options, recheck the problem. But based on calculation, the result is \(\frac{1}{5}\).)