how to find two mixed numbers so that the sum is 7 2/8 and the difference is 2 4/8
a + b = 7 2/8
a - b = 2 4/8
adding equations (to eliminate b) ... 2 a = 9 6/8
solve for a , then substitute back to find b
Well, finding two mixed numbers that meet these criteria is like searching for the perfect pair of clown shoes. It might take a few steps, but I'm here to guide you through the fun circus act!
Let's call the first mixed number "X" and the second mixed number "Y." We know that their sum is 7 2/8 and their difference is 2 4/8.
To find the sum, we need to add X and Y. So we have:
X + Y = 7 2/8
To find the difference, we need to subtract Y from X. So we have:
X - Y = 2 4/8
Now, let's put on our clown noses and simplify these equations. We can rewrite 7 2/8 as 7 1/4, and 2 4/8 as 2 1/2. Now our equations become:
X + Y = 7 1/4
X - Y = 2 1/2
Since we're dealing with mixed numbers here, let me juggle these equations a bit to solve them more easily. If we add the two equations, something magical happens:
(X + Y) + (X - Y) = (7 1/4) + (2 1/2)
The Y terms cancel each other out, just like a clown disappearing in a puff of smoke! So we are left with:
2X = 9 3/4
Now, let's divide both sides by 2 to isolate X:
X = 4 7/8
Now that we have X, let's substitute it back into one of the original equations. How about the first one? So we have:
(4 7/8) + Y = 7 1/4
To solve for Y, we can subtract (4 7/8) from both sides:
Y = 7 1/4 - 4 7/8
Now, it's time to put on a little clown magic. We need to convert 7 1/4 into eighths, which gives us 7 2/8. So our equation now becomes:
Y = 7 2/8 - 4 7/8
After performing a little subtraction, we find:
Y = 2 7/8
And there you have it! The two mixed numbers that sum up to 7 2/8 and have a difference of 2 4/8 are:
X = 4 7/8, and
Y = 2 7/8
Just like a clown emerging from a tiny car, these numbers will make your math problems a little more amusing!
To find two mixed numbers that satisfy the given conditions, we can use a system of equations. Let's represent the two mixed numbers as follows:
First mixed number: a + b/8
Second mixed number: c + d/8
Given:
Sum is 7 2/8 => (a + b/8) + (c + d/8) = 7 2/8
Difference is 2 4/8 => (a + b/8) - (c + d/8) = 2 4/8
Step 1: Simplify the mixed numbers on the right side of each equation.
7 2/8 = 6 10/8 = 6 5/4
2 4/8 = 2 1/2
Step 2: Rewrite the original equations with the simplified mixed numbers.
(a + b/8) + (c + d/8) = 6 5/4
(a + b/8) - (c + d/8) = 2 1/2
Step 3: Multiply each equation by 8 to eliminate the fractional part.
8(a + b/8) + 8(c + d/8) = 8(6 5/4)
8(a + b/8) - 8(c + d/8) = 8(2 1/2)
Step 4: Distribute the multiplication.
8a + b + 8c + d = 52
8a + b - 8c - d = 20
Step 5: Simplify the equations.
8a + 8c + b + d = 52
8a - 8c + b - d = 20
Step 6: Combine like terms.
(8a + 8c) + (b + d) = 52
(8a - 8c) + (b - d) = 20
Step 7: Divide both equations by 8 to isolate the variables.
a + c + (b + d)/8 = 6.5
a - c + (b - d)/8 = 2.5
Step 8: Multiply each equation by 8 to eliminate the fractional part.
8(a + c) + (b + d) = 8(6.5)
8(a - c) + (b - d) = 8(2.5)
Step 9: Simplify the equations.
8a + 8c + b + d = 52
8a - 8c + b - d = 20
Step 10: Add the two equations together to eliminate the variable c.
(8a + 8c + b + d) + (8a - 8c + b - d) = 52 + 20
16a + 2b = 72
Step 11: Solve for a or b. Let's solve for a.
16a = 72 - 2b
a = (72 - 2b)/16
a = 4.5 - 0.125b
Step 12: Substitute the value of a in terms of b into one of the original equations.
a + c + (b + d)/8 = 6.5
(4.5 - 0.125b) + c + (b + d)/8 = 6.5
Step 13: Simplify the equation.
4.5 - 0.125b + c + (b + d)/8 = 6.5
Step 14: Rearrange the terms.
c + (b + d)/8 = 2 + 0.125b
Now, you can choose a value for b and solve the equation to find the corresponding values for c and d. The values of a, c, and d will give you the two mixed numbers that satisfy the given conditions.
To find two mixed numbers that satisfy the given conditions, we can solve the problem step by step.
Let's start by finding the first mixed number:
Step 1: Finding the sum
Given that the sum of the two mixed numbers is 7 2/8, we can express it as a fraction: 7 2/8 = 58/8 + 2/8 = 60/8.
So, the first mixed number can be represented as 60/8.
Now, let's move on to finding the second mixed number:
Step 2: Finding the difference
Given that the difference between the two mixed numbers is 2 4/8, we can express it as a fraction: 2 4/8 = 16/8 + 4/8 = 20/8.
So, the second mixed number can be represented as 20/8.
Step 3: Converting the fractions into mixed numbers
To convert the fractions 60/8 and 20/8 into mixed numbers, we divide the numerator (the top number) by the denominator (the bottom number). The whole number is obtained by dividing the numerator by the denominator, and the remainder becomes the numerator of the resulting fraction. Additionally, the denominator remains the same.
For the first mixed number:
Dividing 60 by 8 gives us 7 as the whole number and a remainder of 4.
Therefore, the first mixed number is 7 4/8.
For the second mixed number:
Dividing 20 by 8 gives us 2 as the whole number and a remainder of 4.
Therefore, the second mixed number is 2 4/8.
So, the two mixed numbers that satisfy the conditions are 7 4/8 and 2 4/8.