The primary
question asks us to determine the number of cups of bleach to add to 5 gallons of water if 4.5
cups is required for 10 gallons of water.
One approach to solving a problem
like this is to see it as a problem of proportions. We can set up a proportion and solve for the
missing amount. The proportion can be written as 4.5:10=x:5 or as an equality involving
ratios:
`4.5/10=x/5` . Order matters; we are comparing cups of bleach to
gallons of water in each example.
To solve we use a property of proportions:
if `a/b=c/d` then ab=cd. (This is known as the means-extremes property. In the given proportion
a and d are the extremes with b and c as the means. Then the product of the means equals the
product of the extremes.)(( Sometimes this is taught as the rule of three; see the
reference.)
Thus 5(4.5)=10x or 22.5=10x. To solve for x we
divide both sides by 10 to get x=2.25. So we need 2.25 (`2 1/4` ) cups of bleach for 5 gallons
of water.
Similar problems include:
(a)
conversions. For example there are 2.54 cm in an inch. Find the number of centimeters in a yard.
Or monetary conversions, like if 1 British pound is worth $1.70 in US currency, how much is 50
pounds worth.
(b) proportions: the median of a triangle is divided into a
ratio of 2:1 by the centroid. If the distance to the centroid from vertex A is 5c m, what is the
length of the segment of the median from the centroid to the side opposite vertex
A.
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