1.a collection of 105 coins consists of 1 peso and 5 pesos. if the total value is 205 pesos, find the number of coins of each denomination in the collection.
2.leslie's wallet contains 20 peso bills and 1 peso coins with a total of 232 pesos. how many each denomination are in her wallet if she has 21 pieces in all?
3. working together, alice and betty can do a certain job in 4 1/3 days. but alice fell ill after 2 days of working and betty finished the job continuing to work alone in 6 3/4 more days. how long it take each to do the job if each of them worked alone?
4. gian and arwin can finish a project in 4 3/2 hours. gian can do it alone in half as much time as arwin can do the job. how long will it take arwin to do the job alone?
5. roma can finish weeding a flower garden in 4 1/2 hours. roma has worked for 2 hours before amor joined her and they finished the job in 2 hours. how long would it take both of them to finish the job working together? how long would it take amor to do the job alone?
Please show your own work for assistance. Someone will critique your work.
Are you allowed to use algebraic methods? They make it a lot easier.
fdhnyu
1. To solve this problem, we can set up a system of equations. Let's denote the number of 1 peso coins as 'x' and the number of 5 pesos coins as 'y'.
The first equation represents the total number of coins: x + y = 105
The second equation represents the total value of the coins: 1x + 5y = 205
We can solve this system of equations using either substitution or elimination. Let's use elimination:
Multiply the first equation by -1 to get: -x - y = -105
Add this modified equation to the second equation: -x - y + 1x + 5y = -105 + 205
Simplifying, we have: 4y = 100
Divide both sides by 4: y = 25
Substitute this value of 'y' into the first equation: x + 25 = 105
Subtract 25 from both sides: x = 80
So, there are 80 1 peso coins and 25 5 pesos coins in the collection.
2. Again, let's denote the number of 20 peso bills as 'x' and the number of 1 peso coins as 'y'.
The first equation represents the total number of bills and coins: x + y = 21
The second equation represents the total value of the bills and coins: 20x + 1y = 232
Let's solve this system of equations using elimination:
Multiply the first equation by 20: 20x + 20y = 420
Subtract this modified equation from the second equation: 20x + 1y - (20x + 20y) = 232 - 420
Simplifying, we have: -19y = -188
Divide both sides by -19: y = 9.89
Since we can't have a fraction of a coin, we need to round down. So, y = 9.
Substitute this value of 'y' into the first equation: x + 9 = 21
Subtract 9 from both sides: x = 12
Therefore, there are 12 twenty peso bills and 9 one peso coins in Leslie's wallet.
3. Let's assume that Alice and Betty's individual rates of work are represented by 'A' and 'B' respectively.
Together, their combined rate of work is 1 job per (4 1/3) days.
So, their combined rate is 1/(13/3) = 3/13 jobs per day.
Let's denote the time Alice worked as 't' days. Therefore, Betty worked for (2 + 6 3/4) - t = (19/4) - t days.
Using the formula Rate x Time = Work, we can calculate the amount of work done by each person.
Alice's work: (A) * t = A*t
Betty's work: (B) * ((19/4) - t) = B*((19/4) - t)
When they work together, their combined work is 1 job.
So, we have the equation: A*t + B*((19/4) - t) = 1
We also know that their combined rate is 3/13 jobs per day.
So, we have the equation: A + B = 3/13
Now, we have two equations with two unknowns. We can solve this system of equations to find the values of A and B.
4. Let's assume that Arwin's time to finish the job alone is 'x' hours. Therefore, Gian's time to finish the job alone is 2x hours.
When they work together, their combined rate is 1 job per (4 3/2) hours.
Let's setup the equation using the formula Rate x Time = Work:
Arwin's work: (1/x) * (4 3/2) = 1
Gian's work: (1/(2x)) * (4 3/2) = 1
Simplifying, we get: 4/(2x) = 1
Cross-multiplying, we have: 4 = 2x
Dividing both sides by 2, we get: 2 = x
Therefore, it will take Arwin 2 hours to do the job alone.
5. Let's assume that Roma's time to finish the job alone is 'x' hours. Therefore, Amor's time to finish the job alone is (x + 4 1/2) hours.
When they work together, their combined rate is 1 job per (2 + 2) = 4 hours.
Using the formula Rate x Time = Work, we can calculate the amount of work done by each person.
Roma's work: (1/x) * (4 1/2) = (9/2x)
Amor's work: (1/(x + 4 1/2)) * (2) = (2/(x + 9/2))
When they work together, their combined work is 1 job.
So, we have the equation: (9/2x) + (2/(x + 9/2)) = 1
Now, we can solve this equation to find the value of x.
We do not do your work for you. Once you have attempted to answer your questions, we will be happy to give you feedback on your work. Although it might require more time and effort, you will learn more if you do your own work. Isn't that why you go to school?
However, I will give you a start.
1. Let x = 1 peso coins and y = 5 peso coins
y = 5x
x + y = 205
Substitute 5x for y in second equation and solve for x. Insert that value into the first equation and solve for y. Check by inserting both values into the second equation.