Solve the following problems:
1. A turtle and a snail are 360 meters apart, and they start to move towards each other at 3 p.m. If the turtle is 11 times as fast as the snail, and they met at 3:40 p.m., find the speed of each.
2. The ratio of the number of dolls Jacky had to the number of dolls Peter had was 5:2 but, after Jacky gave 15 dolls to Peter, they have an equal number of dolls. How many dolls did they have altogether?
3. Jessica has a combined total of 48 nickels and dimes. How many of each does she have if she has $3.25 total?
they traveled 360m in 40 minutes, for a combined speed of 9 m/min
so the snail's speed is 9 * 1/12 = 3/4 m/min
and the turtle's speed is 9 * 11/12 = 33/4 m/min
j/p = 5/2
j-15 = p+15
so,
(5/2 p)-15 = p+15
p = 20, j=50
so they had 70 dolls altogether
n+d = 48
5n+10d = 325
n=31, d=17
To solve these problems, we will use algebraic equations based on the information given.
Problem 1:
Let's assume the speed of the snail is S meters per minute, and the speed of the turtle is T meters per minute.
Given:
Distance = 360 meters
Time = 40 minutes (from 3 p.m. to 3:40 p.m.)
Relative speed = Speed of the snail + Speed of the turtle
Using the formula Distance = Speed * Time, we can write two equations based on the information given:
Equation 1: S * 40 = 360 (since the snail moves for 40 minutes at speed S)
Equation 2: T * 40 = 360 (since the turtle moves for 40 minutes at speed T)
We also know that the turtle is 11 times faster than the snail, so we can write another equation:
Equation 3: T = 11S (since the turtle's speed is 11 times the snail's speed)
Now, we can solve this system of equations to find the values of S and T.
From Equation 3, substitute T = 11S into Equations 1 and 2:
11S * 40 = 360
440S = 360
S = 0.818 meters per minute (rounded to three decimal places)
Using Equation 3, we can find T:
T = 11S
T = 11 * 0.818
T = 9 meters per minute (rounded to one decimal place)
Therefore, the speed of the snail is approximately 0.818 meters per minute, and the speed of the turtle is approximately 9 meters per minute.
Problem 2:
Let's assume the number of dolls Jacky had is J, and the number of dolls Peter had is P.
Given:
Ratio of J to P = 5:2
Jacky gave 15 dolls to Peter, so Jacky has (J - 15) dolls and Peter has (P + 15) dolls.
Based on the information given, we can write an equation using the ratio of dolls:
Equation 1: J/P = 5/2
After Jacky gave 15 dolls to Peter, both of them have an equal number of dolls. So we can write another equation:
Equation 2: J - 15 = P + 15
Now, we can solve this system of equations to find the values of J and P.
Rearrange Equation 1 to express J in terms of P:
J = (5/2)P
Substitute this expression for J into Equation 2:
(5/2)P - 15 = P + 15
Simplify and solve for P:
(5/2)P - P = 15 + 15
3P/2 = 30
3P = 60
P = 20
Now substitute the value of P into Equation 1 to find J:
J = (5/2) * 20
J = 50
Therefore, Jacky originally had 50 dolls and Peter originally had 20 dolls. After Jacky gave 15 dolls to Peter, they have a total of 55 dolls (50 + 20 - 15).
Problem 3:
Let's assume the number of nickels Jessica has is N, and the number of dimes she has is D.
Given:
Total number of coins (nickels and dimes) = 48
Total value of coins = $3.25
Based on the information given, we can write two equations based on the number of coins and their values:
Equation 1: N + D = 48 (since the total number of coins is 48)
Equation 2: 0.05N + 0.10D = 3.25 (since the value of each nickel is $0.05 and the value of each dime is $0.10)
Now, we can solve this system of equations to find the values of N and D.
Rearrange Equation 1 to express N in terms of D:
N = 48 - D
Substitute this expression for N into Equation 2:
0.05(48 - D) + 0.10D = 3.25
Simplify and solve for D:
2.4 - 0.05D + 0.10D = 3.25
0.05D = 3.25 - 2.4
0.05D = 0.85
D = 0.85 / 0.05
D = 17
Now substitute the value of D into Equation 1 to find N:
N = 48 - D
N = 48 - 17
N = 31
Therefore, Jessica has 31 nickels and 17 dimes.
1. Let's first calculate the time it took for them to meet:
The meeting time is 3:40 p.m., which is 40 minutes after they started at 3 p.m.
Now, let's convert the time into hours (since the speed is usually given in units per hour):
40 minutes is equal to 40/60 = 2/3 hours.
Let's say the speed of the snail is S meters per hour. Therefore, the speed of the turtle is 11S meters per hour.
To calculate the distance traveled by the snail in 2/3 hours, we use the formula: distance = speed × time
Distance traveled by the snail = S × (2/3)
To calculate the distance traveled by the turtle in 2/3 hours, we use the same formula:
Distance traveled by the turtle = 11S × (2/3)
Since they met when the sum of their distances is equal to the total distance of 360 meters, we can write the equation:
S × (2/3) + 11S × (2/3) = 360
Simplifying the equation:
(2S + 22S) × (2/3) = 360
(24S) × (2/3) = 360
16S = 360
Dividing both sides by 16:
S = 22.5
Therefore, the speed of the snail is 22.5 meters per hour.
The speed of the turtle is 11 times that of the snail, so the speed of the turtle is:
11 × 22.5 = 247.5 meters per hour.
2. Let's represent the number of dolls Jacky had as 5x, and the number of dolls Peter had as 2x before Jacky gave 15 dolls to Peter.
After Jacky gave 15 dolls to Peter, the number of dolls they both had became equal. Therefore, we can set up the equation:
5x - 15 = 2x + 15
Simplifying the equation:
3x = 30
Dividing both sides by 3:
x = 10
Now, substitute the value of x back into the equation to find the number of dolls they had:
Number of dolls Jacky had = 5x = 5 × 10 = 50 dolls
Number of dolls Peter had = 2x = 2 × 10 = 20 dolls
Altogether, they had 50 + 20 = 70 dolls.
3. Let's denote the number of nickels as N and the number of dimes as D.
We have two pieces of information to form an equation:
1. The total number of coins: N + D = 48
2. The total value of the coins: 0.05N + 0.10D = 3.25
To solve this system of equations, we can start by multiplying the second equation by 100 to simplify the decimal values:
0.05N + 0.10D = 3.25
5N + 10D = 325
Now we can use either the substitution or elimination method to solve for N and D. Let's use the elimination method:
We can multiply the first equation by -5 to match the coefficient of 5N:
-5(N + D) = -5(48)
-5N - 5D = -240
Now we can add the two equations together:
(5N + 10D) + (-5N - 5D) = 325 + (-240)
5D = 85
Dividing both sides by 5:
D = 17
Substituting this value back into the first equation:
N + 17 = 48
N = 31
Therefore, Jessica has 31 nickels and 17 dimes.