Applied Math: Rational Numbers Practice (Solved)

Practice
 

P1. What is 30% of 120?
 

P2. What is 150% of 20?
 

P3. What is 14.5% of 96?
 

P4. Simplify the following expressions:
(a)
(b)
(c)
(d)
(e)
 

P5. Convert the following to a fraction and to a decimal: (a) 15%; (b) 24.36%
 

P6. Convert the following to a decimal and to a percentage. (a) 4/5; (b)
 

P7. A woman’s age is thirteen more than half of 60.
How old is the woman?
 

P8. A patient was given pain medicine at a dosage of 0.22 grams. The patient’s dosage was then increased to 0.80 grams. By how much was the patient’s dosage increased?
 

P9. At a hotel,  of the 100 rooms are occupied today. Yesterday,  of the 100 rooms were occupied. On which day were more of the rooms occupied and by how much more?
 

P10. At a school, 40% of the teachers teach English.
If 20 teachers teach English, how many teachers work at the school?
 

P11. A patient was given blood pressure medicine at a dosage of 2 grams. The patient’s dosage was then decreased to 0.45 grams. By how much was the patient’s dosage decreased?
P12. Two weeks ago,  of the 60 customers at a skate shop were male. Last week,  of the 80 customers were male. During which week were there more male customers?
 

P13. Jane ate lunch at a local restaurant. She ordered a $4.99 appetizer, a $12.50 entrée, and a $1.25 soda. If she wants to tip her server 20%, how much money will she spend in all?
 

P14. According to a survey, about 82% of engineers were highly satisfied with their job. If 145 engineers were surveyed, how many reported that they were highly satisfied?
 

P15. A patient was given 40 mg of a certain medicine.
Later, the patient’s dosage was increased to 45 mg. What was the percent increase in his medication?
P16. Order the following rational numbers from least to greatest: 0.55, 17%, , , , 3.
P17. Order the following rational numbers from greatest to least: 0.3, 27%, , , . 4.5
 

P18. Perform the following multiplication. Write each answer as a mixed number.
(a)
(b)
(c)
 

P19. Suppose you are making doughnuts and you want to triple the recipe you have. If the following list is the original amounts for the ingredients, what would be the amounts for the tripled recipe?
 

	cup	Flour
	tsp	Baking powder
	tsp	Salt
	cup	Sugar
	Tbsp	Butter
	large	Eggs
	tsp	Vanilla extract
	cup	Sour cream

 
Practice Solutions:

P1. The word of indicates multiplication, so
30% of 120 is found by multiplying 120 by 30%. Change 30% to a decimal, then multiply:
 

P2. The word of indicates multiplication, so
150% of 20 is found by multiplying 20 by 150%. Change 150% to a decimal, then multiply:
 

P3. Change 14.5% to a decimal before multiplying. .
 

P4. Follow the order of operations and utilize properties of fractions to solve each:
(a)
Rewrite the problem as a multiplication problem: . Make sure the fraction is reduced to lowest terms. Both 14 and 20 can be divided by 2.

 

(b) The denominators of  and  are 8 and 16, respectively. The lowest common denominator of 8 and 16 is 16 because 16 is the least common multiple of 8 and 16. Convert the first fraction to its equivalent with the newly found common denominator of 16: . Now that the fractions have the same denominator, you can subtract them.

 

(c) When simplifying expressions, first perform operations within groups. Within the set of parentheses are multiplication and subtraction operations. Perform the multiplication first to get . Then, subtract two to obtain  Finally, perform addition from left to right:

 

(d) First, evaluate the terms in the parentheses  using order of operations. , and .
Next, rewrite the problem: . Finally, add and subtract from left to right: ; .
The answer is .
 

(e) First, simplify within the parentheses, then change the fraction to a decimal and perform addition from left to right:

 
P5. (a) 15% can be written as . Both 15 and 100 can be divided by 5:
When converting from a percentage to a decimal, drop the percent sign and move the decimal point two places to the left: .
(b) 24.36% written as a fraction is , or , which reduces to
. 24.36% written as a decimal is 0.2436. Recall that dividing by 100 moves the decimal two places to the left.
P6. (a) Recall that in the decimal system the first decimal place is one tenth:
Percent means “per hundred.”
(b) The mixed number  has a whole number and a fractional part. The fractional part  can be written as a decimal by dividing 5 into 2, which gives 0.4. Adding the whole to the part gives 3.4.
To find the equivalent percentage, multiply the decimal by 100. . Notice that this percentage is greater than 100%. This makes sense because the original mixed number  is greater than 1.
 

P7. “More than” indicates addition, and “of” indicates multiplication. The expression can be written as . So the woman’s age is equal to . The woman is 43 years old.
 

P8. The first step is to determine what operation
(addition, subtraction, multiplication, or division) the problem requires.
Notice the keywords and phrases “by how much” and “increased.” “Increased” means that you go from a smaller amount to a larger amount. This change can be found by subtracting the smaller amount from the larger amount:  grams.
Remember to line up the decimal when subtracting:

 
P9. First, find the number of rooms occupied each day. To do so, multiply the fraction of rooms occupied by the number of rooms available:

 
The difference in the number of rooms occupied is:
 

P10. To answer this problem, first think about the number of teachers that work at the school. Will it be more or less than the number of teachers who work in a specific department such as English? More teachers work at the school, so the number you find to answer this question will be greater than 20.

40% of the teachers are English teachers. “Of” indicates multiplication, and words like “is” and “are” indicate equivalence. Translating the problem into a mathematical sentence gives , where t represents the total number of teachers. Solving for t gives . Fifty teachers work at the school.
P11. The decrease is represented by the difference between the two amounts:

Remember to line up the decimal point before subtracting.

 
P12. First, you need to find the number of male customers that were in the skate shop each week. You are given this amount in terms of fractions. To find the actual number of male customers, multiply the fraction of male customers by the number of customers in the store.

 
The number of male customers was the same both weeks.
 

P13. To find total amount, first find the sum of the items she ordered from the menu and then add
20% of this sum to the total.



 

P14. 82% of 145 is . Because you can’t have 0.9 of a person, we must round up to say that 119 engineers reported that they were highly satisfied with their jobs.
 

P15. To find the percent increase, first compare the original and increased amounts. The original amount was 40 mg, and the increased amount is 45 mg, so the dosage of medication was increased by 5 mg (). Note, however, that the question asks not by how much the dosage increased but by what percentage it increased.

 
P16. Recall that the term rational simply means that the number can be expressed as a ratio or fraction. Notice that each of the numbers in the problem can be written as a decimal or an integer:

 So, the answer is 17%, , 0.55, 3, , .
 

P17. Converting all the numbers to integers and decimals makes it easier to compare the values:

 
So, the answer is , , 4.5, 0.3, 27%, .
 

P18. For each, convert improper fractions, adjust to a common denominator, perform the operations, and then simplify:
 

(a)
Sometimes, you can skip converting the denominator and just distribute the multiplication.

 
(b)

 
(c)

 
P19. Fortunately, some of the amounts are duplicated, so we do not need to figure out every amount.

 
So, the result for the triple recipe is:

	cup	Flour
	tsp	Baking powder
	tsp	Salt
	cup	Sugar
	Tbsp	Butter
	large	Eggs
	tsp	Vanilla extract
	cup	Sour cream