1. find the two consecutive number such that five-eights of the smaller number equals three-fifths of the larger number.
2. separate 221 into two parts such that their quotient is 8/9.
thanks...
5/8 x = 3/5 (x+1)
x/(221-x) = 8/9
It also helps to know that 221=17*13
1. Let's assume the smaller consecutive number as x.
According to the given condition, five-eighths of the smaller number is equal to three-fifths of the larger number.
So we can write the equation as (5/8)x = (3/5)(x+1).
Let's solve this equation step-by-step:
Step 1: Simplify the equation.
(5/8)x = (3/5)(x+1)
Multiply both sides by 8 and 5 to eliminate the fractions:
5(5/8)x = 8(3/5)(x+1)
25x/8 = 24/5(x+1)
Step 2: Cross multiply and simplify.
(25x/8) * (5/3) = (24/5) * (x+1)
25x/24 = 8(x+1)
Step 3: Distribute and simplify.
(25x*5) / (24*3) = 8x + 8
125x/72 = 8x + 8
Step 4: Subtract 8x from both sides.
125x/72 - 8x = 8
Step 5: Simplify the left side.
(125x - 576x)/72 = 8
Step 6: Combine like terms.
-451x/72 = 8
Step 7: Multiply both sides by 72.
(-451x/72) * 72 = 8 * 72
Step 8: Simplify.
-451x = 576
Step 9: Divide both sides by -451.
x = -576/451
So, the two consecutive numbers are approximately -576/451 and (-576/451) + 1.
2. Let's assume the two parts of 221 as x and y.
According to the given condition, their quotient is 8/9.
So we can write the equation as x/y = 8/9.
Let's solve this equation step-by-step:
Step 1: Simplify the equation.
x/y = 8/9
Step 2: Cross multiply.
9x = 8y
Step 3: Solve for one variable in terms of the other.
x = (8y)/9
Step 4: Substitute into the original equation.
(8y)/9 = y
Step 5: Multiply both sides by 9 to eliminate the fraction.
8y = 9y
Step 6: Subtract 8y from both sides.
y = 0
Step 7: Substitute the value of y into the equation.
x = (8*0)/9
x = 0
So, the two parts of 221 that satisfy the given condition are 0 and 0.
1. To find two consecutive numbers, let's denote the smaller number as x and the larger number as x+1. We can set up the equation based on the given information:
(5/8) * x = (3/5) * (x+1)
To solve for x, we can cross multiply:
5 * (x+1) = 8 * (3/5) * x
5x + 5 = (24/5) * x
To get rid of the fractions, we can multiply both sides of the equation by 5:
5 * (5x + 5) = 5 * [(24/5) * x]
25x + 25 = 24x
Next, we can subtract 24x from both sides:
25x - 24x + 25 = 0
x + 25 = 0
Lastly, we subtract 25 from both sides to solve for x:
x = -25
So, the smaller consecutive number is -25, and the larger consecutive number is -25+1 = -24.
2. To separate 221 into two parts such that their quotient is 8/9, let's denote the two parts as x and y. We can set up the equation based on the given information:
x / y = 8/9
To find the values of x and y, we can cross multiply:
9 * x = 8 * y
To simplify the equation, we need to express 221 as the sum of x and y. However, since there are multiple solutions, let's assume that x is the larger part and y is the smaller part:
x = y + k
where k represents the difference between the two parts.
Substituting the expression for x into the equation:
9 * (y + k) = 8 * y
Expanding the equation:
9y + 9k = 8y
To solve for k, we can subtract 8y from both sides:
9y - 8y + 9k = 0
y + 9k = 0
Now, to find a suitable value for k, we need to choose a value that satisfies the equation. Let's try k = 9:
y + 9(9) = 0
y + 81 = 0
y = -81
We have found a solution where y is -81, so to find x, we can substitute y back into the equation for x:
x = -81 + k
x = -81 + 9
x = -72
Therefore, the two parts that separate 221 into a quotient of 8/9 are -72 and -81.