The twenty six coins in my pocket are all dollar coins and quarters, and they add up to seventeen dollars in value. How many dollar coins and quarters are in my pocket?
Let's assume the number of dollar coins in the pocket is d and the number of quarters is q.
We can create a system of equations based on the given information:
Equation 1: d + q = 26 (since there are twenty-six coins in total)
Equation 2: 1d + 0.25q = 17 (since the total value of the coins is seventeen dollars)
From Equation 1, we can isolate d: d = 26 - q
Substituting this value of d into Equation 2:
1(26 - q) + 0.25q = 17
Expanding the equation:
26 - q + 0.25q = 17
Combine the like terms:
26 - 0.75q = 17
Subtract 26 from both sides:
-0.75q = 17 - 26
-0.75q = -9
Divide by -0.75:
q = -9 / -0.75
q = 12
Now that we know q is 12, we can substitute this value back into Equation 1:
d + 12 = 26
d = 26 - 12
d = 14
Therefore, there are 14 dollar coins and 12 quarters in the pocket.
Jen ran 1 2/3 miles in 1/4 of an hour, and Max ran 1 1/2 miles in 1/5 of an hour. What is the average speed of both Jen and Max? Who runs faster?
To find the average speed of Jen and Max, we need to divide the distance each person ran by the time it took to run that distance.
Jen's speed can be calculated by dividing the distance, 1 2/3 miles, by the time, 1/4 hour:
Jen's Speed = (1 2/3 miles) / (1/4 hour)
Jen's Speed = (5/3 miles) / (1/4 hour)
To divide fractions, we need to multiply by the reciprocal of the divisor:
Jen's Speed = (5/3 miles) * (4/1 hour)
Jen's Speed = 20/3 miles per hour
Max's speed can be calculated in the same way:
Max's Speed = (1 1/2 miles) / (1/5 hour)
Max's Speed = (3/2 miles) / (1/5 hour)
Max's Speed = (3/2 miles) * (5/1 hour)
Max's Speed = 15/2 miles per hour
To compare their speeds, we can simplify their speeds by finding a common denominator:
Jen's Speed = (20/3) miles per hour
Max's Speed = (15/2) miles per hour
Finding a common denominator of 6:
Jen's Speed = (40/6) miles per hour
Max's Speed = (45/6) miles per hour
Now we can see that Max's speed is greater than Jen's speed, as 45/6 > 40/6.
Therefore, Max runs faster than Jen, with an average speed of 45/6 miles per hour.
I thought of a number, added 4 5/7 to it, and got a number equal to the original number multiplied by 12. What was the original number?
Let's say the original number is represented by "x".
According to the given information, when we add 4 5/7 to the original number, we get a number that is equal to the original number multiplied by 12.
Mathematically, we can represent this as:
x + 4 5/7 = 12x
To make it easier to work with, let's convert 4 5/7 into an improper fraction:
4 5/7 = (4 * 7 + 5) / 7 = 33/7
Now our equation becomes:
x + 33/7 = 12x
To get rid of the fraction, we'll multiply both sides of the equation by 7:
7x + 33 = 84x
Next, we'll isolate the variables by subtracting 7x from both sides of the equation:
7x - 7x + 33 = 84x - 7x
33 = 77x
Now, divide both sides of the equation by 77:
33/77 = 77x/77
Simplifying the fraction:
3/7 = x
Therefore, the original number is 3/7.
Adele bought 2 3/5 lb of grapes and Taylor Swift bought 1/2 less lb than Adele. How many pounds of grapes did they buy in all?
Adele bought 2 3/5 pounds of grapes.
To find out how much Taylor Swift bought, we need to subtract 1/2 from Adele's amount:
2 3/5 - 1/2
To subtract fractions, we need a common denominator. The common denominator of 5 and 2 is 10.
Converting 2 3/5 to an improper fraction:
2 3/5 = (2 * 5 + 3)/5 = 13/5
Converting 1/2 to fifths by multiplying the numerator and denominator by 5:
1/2 = (1 * 5)/(2 * 5) = 5/10
Now, let's subtract:
13/5 - 5/10
To subtract fractions, we need a common denominator, which is 10:
13/5 - 5/10 = (26/10) - (5/10) = 21/10
Therefore, Taylor Swift bought 21/10 pounds of grapes.
To find out how many pounds of grapes they bought in all, we need to add the amounts:
2 3/5 + 21/10
To add fractions, we need a common denominator, which is 10:
2 3/5 + 21/10 = (2 * 10 + 3)/10 + 21/10 = 23/10 + 21/10
Now, we can add the fractions:
23/10 + 21/10 = (23 + 21)/10 = 44/10
To simplify this fraction, we can divide the numerator and denominator by their greatest common divisor, which is 2:
44/10 = (2 * 22)/(2 * 5) = 22/5
Therefore, Adele and Taylor Swift bought a total of 22/5 pounds of grapes.
A 35 inch stick is cut up into two pieces. If the ratio of their lengths is 2:3, then how long is each piece?
To solve this problem, we can use a system of linear equations. Let's represent the number of dollar coins as "d" and the number of quarters as "q".
We can set up two equations based on the given information:
1. The total number of coins: d + q = 26
2. The total value of the coins: $1(d) + $0.25(q) = $17
Now, we can solve this system of equations to find the values of d and q.
To solve the first equation, we can isolate d:
d = 26 - q
Substitute this value of d into the second equation:
$1(26 - q) + $0.25(q) = $17
Simplify the equation:
$26 - $1q + $0.25q = $17
Combine like terms:
$0.75q = $17 - $26
$0.75q = -$9
Divide both sides of the equation by $0.75 to solve for q:
q = -9 / 0.75
q = -12
Uh-oh! It looks like the solution resulted in a negative number of quarters. This means that the given information is not consistent, and there may be a mistake.
Please double-check the information provided and make sure all the given numbers are correct.