Jim needs 4 1/2 cups of bleach for every ten gallons of water for cleaning. How many cups of bleach should he use if he is making 5 gallons?    

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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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