Word problems with ratios and fractions often require us to think carefully about the distinction between a ratio and a fraction. Remember that a fraction MUST compare to the whole.
Example: Forty of the ninety farm animals are cows. The rest are chickens.
This must be a part-to-whole expression. The denominator must be the total number of farm animals. We have the fraction
The cows are four-ninths of the farm animals.
Further, the part-to-whole ratio is 4 : 9 (cows to animals). Also, the whole-to-part ratio is 9 : 4 (animals to cows).
This must be a part-to-part expression. The ratio of 40:50 simplifies to 4:5 or 
.
The ratio of cows to chickens is 4 to 5.
Note that the sum of the two values represents the whole, e.g., 40 + 50 = 90 and
4 + 5 = 9. So, the part to whole ratios are 4:9 (cows to animals) and 5:9 (chickens to animals). Also, the whole to part ratios are 9:4 (animals to cows) and 9:5 (animals to chickens).
A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator the smaller the fraction.
Tolstoy, Count Lev Nikolgevich (1828-1920)
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Date last modified: June 11, 2010.
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