Lincoln and Michael had some money in the ratio 7:5. Lincoln spent all his money on 3 similar toothbrushes and 5 similar towels. Michael spent all his money on 5 similar toothbrushes and 3 similar towels. Express the cost of a toothbrush as a ratio to the cost of towel.
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If
x = cost of toothbrush
y = cost of towel
(3x+5y)/(5x+3y) = 7/5
5(3x+5y) = 7(5x+3y)
15x + 25y = 35x + 21y
20x = 4y
x/y = 4/20 = 1/5
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To solve this problem, we need to figure out the amount of money each person spent on toothbrushes and towels.
Let's assume that Lincoln has $7x and Michael has $5x. Since they spent all their money, we can equate their total expenses to their total money.
Lincoln's expenses:
Lincoln spent all his money on 3 similar toothbrushes and 5 similar towels. Let's say the cost of a toothbrush is $t and the cost of a towel is $w.
The total cost of toothbrushes = $t * 3
The total cost of towels = $w * 5
So, we can write the equation:
$t * 3 + $w * 5 = $7x
Michael's expenses:
Michael spent all his money on 5 similar toothbrushes and 3 similar towels. Let's say the cost of a toothbrush is still $t and the cost of a towel is still $w.
The total cost of toothbrushes = $t * 5
The total cost of towels = $w * 3
So, we can write the equation:
$t * 5 + $w * 3 = $5x
Now, we have two equations with two unknowns ($t and $w). We can solve this system of equations to find the cost of a toothbrush as a ratio to the cost of a towel.
To solve these equations, we can rearrange them and use the method of substitution or elimination to find the values of $t and $w. Once we have those values, we can express the cost of a toothbrush as a ratio to the cost of a towel by dividing the cost of a toothbrush by the cost of a towel.
Note: The specific values of $t and $w will depend on the values of x that satisfy the equations.