Write the fraction A. The divisors of 1001 are 7, 11, and 13. Are any of these divisors of 1540? Yes, 7 divides 1540 and 11 divides 1540.
Q. Determine if the fraction
A. Yes. The numerator is divisible by 11 and 13. The denominator is divisible by 9 and 37. They don’t share any common factors, so this fraction is in its lowest terms.
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12 Determine if the fraction is in lowest terms:
13 Determine if the fraction is in lowest terms:
14 Determine if the fraction is in lowest terms:
Making Proportional Statements
A proportion is an equation with two fractions equal to one another. Proportions have some wonderful properties that make them useful for solving problems — especially when you’re comparing one quantity to another or one percentage to another.
Given the proportion
. (The cross-products form an equation.)
. (The “flip” is an equation.)
. (You can reduce either fraction vertically.)
. (You can reduce the numerator or denominator horizontally.)
A. The numerator and denominator in the fraction on the left have a common factor of 6. Multiply each by
Q. If Agnes can type 60 words per minute, how long will it take her to type a manuscript containing 4,020 words (if she can keep typing at the same rate)?
A. Set up a proportion with words in the two numerators and the corresponding number of minutes in the denominators:
Divide both numerators by 60 and then cross-multiply to solve for x.
It will take her 67 minutes — just over an hour.
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16 Solve for x:
17 Solve for x:
18 Solve for x:
19 A recipe calls for 2 teaspoons of cinnamon and 4 cups of flour. You need to increase the flour to 6 cups. To keep the ingredients proportional, how many teaspoons of cinnamon should you use?
20 A factory produces two faulty tablets for every 500 tablets it produces. How many faulty tablets would you expect to find in a shipment of 1,250?
Finding Common Denominators
Before you can add or subtract fractions, you need to find a common denominator for them. Ideally, that common denominator is the least common multiple (the smallest number that each of the denominators can divide into without a remainder). A method of last resort, though, is to multiply the denominators together. Doing so gives you a number that the denominators divide evenly. You may have to work with larger numbers using this method, but you can always reduce the fractions at the end.
And then, there’s the box method. This method is especially helpful when you have three or more fractions to deal with — and they have relatively large denominators.
Creating common denominators from multiples of factors
Common denominators (the same numbers in the denominators) are necessary for adding, subtracting, and comparing fractions. Carefully selected fractions that are equal to the number 1 are used to create common denominators because multiplying by 1 doesn’t change a number’s value.
Follow these steps to find a common denominator for two fractions and write the equivalent fractions:
1 Find the least common multiple of the two denominators — the smallest number that both denominators divide evenly.First, look to see if you can determine the common multiple by simple observation; you may know some multiples of the two numbers. If you find the common multiple by observation, go directly to Step 4. (Do not pass Go; do not collect $200.)
2 If the common multiple isn’t easily determined, start your search by choosing the larger denominator.Check to see if the smaller denominator divides the larger one evenly. If it does, then you’ve found the common denominator. Go to Step 4.
3 Check consecutive multiples of the larger denominator until you find one that the smaller one divides.That’s your common denominator.
4 When you find a common denominator, rewrite both fractions as equivalent fractions with that denominator.Multiply both numerator and denominator of each fraction by the equivalent of 1 that creates fractions with the common denominator.
