Gemma recently rode her bike to visit her friend who lives 12 miles away. on her way there her average speed was 7 miles per hour faster that on her way home. if ge...

Distance, Rate, and Time Applications Sergio rode his bike $4 \mathrm{mi}$. Then he got a flat tire and had to walk back 4 mi. It took him 1 hr longer to walk than it did to ride. If his rate walking was 9 mph less than his rate riding, find the two rates.

We need to calculate the rate off Jessica the radar Jessica is nothing but the distance covered by hope from her home to her friend's home. In 10 minutes she has covered 1.4 minus 0.9 equals 0.5 mice. That is now we need to calculate the late that is 0.5 by 12 minus two, which is nothing but zero point fight by then. So this is nothing but zero points. It'll find this is my spare minutes. Now we need to calculate the rate in my spare our betters. We need to multiply this by one by 60 which is nothing but three. So the answer this three miles but over

All right. How's it going? Let's do some slope stuff. So look at this question. We got Jessica. She's walking home after two minutes. Was about 1.4 miles from her home. After about 12 minutes, she is 0.9 miles from her home. And so it might get a little tricky. Um, if you start thinking about like, Oh, her house is right here and she's over here. But now she's over here after some unit of time, like, let's just and something like that thing about this and X and Y coordinates. Okay, so let's say let's cause we want to find, um, MPH, right? So we find slope that's rise overrun. So we want to find her miles for our Okay, So rise. That's why over X. So let's call why her miles? And let's call X her time. Right. So the first point we get is that two minutes, she has 1.4 miles. Cool. Not the next point we get. It's 12 minutes at 0.9 miles. Okay, so now if you want a fair at the slope of this, you take Let's call this, um, Will you just This is one, and this is to So we're gonna take why? To minus y one over x two minus x one. So that's going to be 0.4 minus 1.9, divided by 12 minus two, which is gonna give us negative 0.5 over 10. All right, um, this could also be rewritten as negative 1/2 over 10 or negative 1/20. So might seem like she's decreasing at a rate of one mile for every 20 minutes. But that's really just her rate as she approaches home. So she's moving at a rate of one mile for every 20 minutes. Or if you want to think about this in terms of MPH, we know that 20 Minutes is just 1/3 of a mob third of an hour, so it's one divided by 1/3. So her mild for our right now is three three MPH. Because she's traveling one mile every 20 minutes

As part of a conditioning program. We see that a jogger Iran, eight miles at the same time. That a cyclist road for 20 of miles? Yes, we have a jogger jogger who ran. And we also have the cyclist who wrote. Then we're also told that the rate of the cyclists was 12 MPH faster. Then the rate of the jogger faster, then the rate of the jogger. Okay, so we're asked to find the rates of the jogger in terms of the cyclist. Okay, so now that we have this and all of our information, we're gonna look at our which is going to designate the jogger speed, huh? And we know that speed is going to equal our distance over time, but we want to look at time so we rearrange our formula to be time equals distance over speed. Hi. So now that we have that, we can create a formula for us to be able to solve. So for the 1st 1 we're gonna dio they ate miles of the jogger over our and we're gonna set that equal to the 20 miles a cyclist did over R plus 12 cause they're going 12 MPH faster. And the reason we set them equally to each other is because they're doing it simultaneously. One just was able to move quicker and get a lot more miles, and but they're still doing it in the same time. So now that we have this, we need to multiply and find the lowest common denominator. So here we find that it's going to be our times are plus the 12. So are our plus 12 and then we're gonna go through and multiply that out. So when we do that, we get are eight r plus 96 equals 20 are okay. And how we got that was it was eight times the r plus 12 equals 20 are And we went through and multiplied out. So eight times, Ari, times 12. So now that we have that, we need to get everything over to the left hand side. So we're gonna have are No. 20 are 20 r minus eight are is going to equal or 96. So then we would have combined 12 r equals 96. So then we would divide by 12. So we have our is equal to eight So we know that the speed of the jogger is going to be eight MPH when the speed of the cyclist is 20 MPH and we can double check ourselves on that because we were told that the cyclist was going 12 MPH faster than the jogger. So 20 is going to be 12 miles faster MPH, faster than eight. So be the joggers eight and speed of cyclist, 20 MPH.

All right. So we have two sisters that are biking and one sister can bike three MPH faster than the other sister. They both bike 36 miles and it takes the fastest sister one hour or less. So what are their speeds? So that means I'm gonna be using distance equals rate times time. And if we want to use our ratio that means our time is going to be equal to our distance divided by our rate. So we are first sister, she went to 36 miles at a rate of X plus three since she was three miles faster than her sister. Which means her time would be 36 divided by the quantity of X plus three. Her sister on the other hand of course went 36 miles also but a rate of X. And so her time is going to be 36 over X. Now we're going to set up an equation so we're gonna have 36 over X plus three minus r. 36 over X. Because the time difference was one hour so that should equal one hour. We then need to multiply by our least common multiple for racial um the bottom of the denominator which is X plus three and X. So we multiply everything by X. And X plus three. When we do that we're gonna have 36 times are X times X plus three All over X. Plus three. And that means they're X plus threes on top and bottom will cancel out minus. We're gonna have our 36 times are X times X plus three. All over X. Which means our exes are going to cancel out and that's going to be equal to one times anything. Is anything. Which means we're gonna have X times are X plus three. Now we need to go ahead and run to simplify. So we have 36 X minus over here we're gonna have 36 times the quantity X plus three. And that's going to be equal to R. X. Times are X plus three now of 36 X. Can't do anything with that. We have to distribute negative 36. So that means you're gonna have negative 36 times X. Which is negative 36 X. And then you're going to have negative 36 times three which is negative 108 And that's going to be equal to X squared because X. Times X. And X times three which is plus three X. Our 36 axis cancel out. So what we're left with then is X squared plus our three eggs. I'm gonna bring our negative +108 over to the other side. So that's gonna be plus one oh eight equals zero. Now we want to go ahead and want to factor this to solve. So that means we're gonna have two sets of parentheses and that should be equal to zero. We know we have X and X are positive and positive means we have plus and plus. So now we're looking for factors of +108 The add +23 This is not possible since there are no factors one oh eight that add to three. So that means we have no answers. Our solution. This cannot be solved.