Question

vet clinic has a vet technician to the of 5 to 2. the total number of vet technicians is 42. how many vet techs per employee? 10 jason practices playing the drums for 25 minutes every 2 days. how many minutes will he practice after 5 weeks?

Explanation:

First Sub - Question (Vet Clinic Problem)

Step1: Determine the ratio

The ratio of vet technicians to vet assistants is \(5:2\), and the number of vet technicians is 42. Let the number of vet assistants be \(x\). So we have the proportion \(\frac{5}{2}=\frac{42}{x}\)? Wait, no, maybe I misread. Wait, maybe the ratio is vet technicians to something else. Wait, the text is a bit unclear, but assuming the ratio of vet technicians to vet assistants is \(5:2\) and we know the number of vet technicians is 42, we can set up a proportion. Wait, maybe it's the other way. Wait, maybe the ratio of vet assistants to vet technicians is \(5:2\)? No, let's re - examine. The original text: "Vet Clinic has a vet technician to... the of 5 to 2. The total number of vet... how many vet...". Wait, maybe the ratio of vet technicians to vet assistants is \(5:2\), and we know the number of vet technicians is, say, if the number of vet technicians is 5 parts and vet assistants is 2 parts. Wait, maybe the number of vet technicians is 42, and we need to find the number of vet assistants. So cross - multiply: \(5x = 42\times2\), \(x=\frac{84}{5}=16.8\)? That doesn't make sense. Maybe the ratio is vet assistants to vet technicians is \(5:2\). So if vet technicians are 2 parts and vet assistants are 5 parts. If vet technicians are 42, then 1 part is \(42\div2 = 21\), and vet assistants are \(21\times5 = 105\). But the question is a bit unclear. Wait, maybe the first sub - question is: A vet clinic has a ratio of vet technicians to vet assistants of 5:2. The number of vet technicians is 42. How many vet assistants are there?

Step1: Define the ratio

Let the number of vet technicians be \(5y\) and vet assistants be \(2y\). We know \(5y = 42\)? No, that's not right. Wait, maybe the ratio is vet assistants to vet technicians is \(5:2\). So vet technicians \(= 2k\), vet assistants \(= 5k\). If vet technicians \(= 42\), then \(2k=42\), \(k = 21\), so vet assistants \(=5\times21 = 105\).

Second Sub - Question (Jason's Drumming Practice)

Step1: Calculate days in 5 weeks

There are 7 days in a week, so in 5 weeks, the number of days is \(5\times7=35\) days.

Step2: Determine practice frequency

Jason practices every 2 days. So the number of times he practices in 35 days is \(\lfloor\frac{35}{2}
floor = 17\) (if we consider whole practice sessions) or if we consider that every 2 days he practices once, the number of intervals is \(\frac{35}{2}=17.5\), but we take the number of times he practices. But the question says "how many minutes will he practice after 5 weeks?". He practices 25 minutes every 2 days.

Step3: Calculate total practice time

First, find the number of practice sessions. In 5 weeks (35 days), the number of 2 - day intervals is \(\frac{35}{2}=17.5\). But since he practices every 2 days, the number of times he practices is 17 (if we start on day 1, he practices on day 1, 3, 5,..., 35 which is 18 times? Wait, 35 divided by 2 is 17.5, so the number of practice sessions is 18 (because on day 1, 3,..., 35 (which is \(1 + 2\times17=35\))). Wait, \(n\)th term of an arithmetic sequence: \(a_n=a_1+(n - 1)d\), \(a_1 = 1\), \(d = 2\), \(a_n=35\). So \(35=1+(n - 1)\times2\), \(34=(n - 1)\times2\), \(n - 1 = 17\), \(n = 18\).

Step4: Calculate total minutes

He practices 25 minutes each time, so total minutes \(=18\times25 = 450\) minutes.

First Sub - Question Answer (assuming vet assistants calculation)

If the ratio of vet assistants to vet technicians is \(5:2\) and vet technicians \( = 42\), then number of vet assistants \(=105\).

Second Sub - Question Answer

Step1: Days in 5 weeks

\(5\) weeks have \(5\times7 = 35\) days.

Step2: Number of practice sessions

Every 2 days he practices, so number of practice sessions \(=\frac{35}{2}=17.5\), but since he starts on day 1, the number of sessions is 18 (as calculated above).

Step3: Total practice time

Total time \(=18\times25=450\) minutes.

Answer:

For the Jason's drumming practice: 450 minutes.

(Note: The first sub - question has unclear text, but the second one is solvable as above. If the first sub - question's ratio is vet technicians to vet assistants \(5:2\) and vet technicians are, say, 10 (a more reasonable number), then vet assistants would be 4. But with the given 42, the calculation gives a non - integer if the ratio is vet technicians to vet assistants \(5:2\). So there might be a misprint in the problem. However, for the second sub - question, the answer is 450 minutes.)