Basic Math Review
Courtesy of the
Student Success Center
Revised 2016 Sharon Stiehm Computation/Arithmetic Natural Numbers
These are the numbers used for counting, sometimes called “Counting Numbers.” They are … 1, 2, 3, 4, 5, 6, 7, 8, 9, 10… There is no largest natural number and the smallest natural number is one (1). Whole Numbers
To answer questions such as, “How many” and “How much,” we generally use whole numbers. The set or collection of whole numbers is 0, 1, 2, 3, 4, 5, 6, 7, … There is no largest whole number, and the smallest whole number is zero (0). Integers
Consist of all Natural Numbers, Whole Numbers, and Negative Numbers. These numbers can best be illustrated on a “number line.” For each Natural Number, 1, 2, 3, 4, 5, …, there is an opposite number to the left of zero, -1, -2, -3, -4, -5, and so on. Negative numbers are considered to be the “opposites” of positive numbers. Rational Numbers
This is a larger number system, which includes the number systems mentioned above and “quotients of integers with nonzero divisors.” Quotients of integers are commonly expressed as “Fractions and Decimals.” Examples of these are… 2/3, -2/3, 7/1, -0.17, 0/8, -8.75, .3334
Every rational number has a point on the number line. However, there are some points on the line for which there is no rational number. These points correspond to what is called “irrational numbers.” Fractions
Fractions are a portion of a whole number and they consist of a numerator, denominator, and fraction bar. In the fraction ½, the (1) one is the numerator (2) two is the denominator (/ or --) is the fraction bar that separates the numerator and denominator. Fractions are another way of showing division. One-half (1/2) can be expressed as 1 ÷ 2. When evaluating this expression we get the decimal five-tenths (.5).
Revised 2016 Sharon Stiehm Decimals
Decimals are fractions whose denominators are powers of ten (10). The number of decimal places equals the number of zeros in the denominator: 1/4 or 25/100 in decimal form is .25 ¾ or 75/100 in decimal form is .75 When adding or subtracting decimal numbers, place the numbers in a column with the decimal points lined up vertically. Percentage
“Of all wood harvested, 35% is used for paper production.” This means for every 100 tons of wood harvested, 35 tons is used to produce paper. 35% is also considered a ratio of 35 to 100, or 35:100. Using a percentage is a more convenient way of showing this relationship. A percentage is changed to a decimal number by dropping the percent sign and moving the decimal two places to the left. (Example: 37.5% = .375 and 45% = .45) Irrational Numbers
As mentioned before, irrational numbers represent points on the number line for which there is no rational number. Examples of irrational numbers are… π (read as “pi”) and √2 (square root of “2”). π is used in finding the area and circumference of a circle, the approximate number for π is 3.141592654… √2 is the length of the diagonal of a square with sides of length one (1). There is no rational number that can be multiplied by itself to get 2. 1.4 is an approximation of √2 because (1.4)2 = 1.96 1.4142 is an even better approximation because (1.4142)2 = 1.99996164 Real Numbers
Real Numbers are the set of all numbers corresponding to points on the number line and include… Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Applications
An application is one of the main uses of mathematics. It is how we use our math skills in everyday life. (The previous explanation for percentage is a good example). Problem Solving
Problem solving is the other main use of mathematics. Look at a situation and try to translate the problem into mathematical terms.
Revised 2016 Sharon Stiehm
Five Steps for Problem Solving 1. Familiarize yourself with the situation. If the problem is described in words, read it carefully. Draw a picture whenever possible. Choose a letter like a, b, c, n, x, y, z, etc. to represent the unknown quantity. In math, we call this letter the “variable.” 2. Translate the problem into an equation. 3. Solve for the variable. 4. Check the answer in the original word problem. 5. State the answer to the problem with appropriate units. Order of Operations
Rules for “Order of Operations”:
1. Begin with the innermost expression, do all calculations within parentheses ( ), brackets [ ], and braces { } before doing expressions outside these symbols. 2. Evaluate all exponential expressions. 3. Do all multiplications and divisions in order from left to right. 4. Do all additions and subtractions in order from left to right.
Revised 2016 Sharon Stiehm Whole Number Problems
1. True or False: -1, 2, 4, -3, 0, and 12 are all whole numbers.
2. True or False: 3, 4, 7, 78, and 1,000,000 are all whole numbers.
3. What digit is in the ones place? 2356
4. What digit is in the tens place? 2356
5. Write a sentence in which the number 250,000,000 is used.
6. Connor is 6’4” tall and weighs 170 lbs. If Connor wants to weigh 190 lbs. by the next basketball season, how many pounds per month does he need to gain? The basketball season starts in 4 months.
7. What is the sum of these numbers: 123, 556, 2, and 2?
8. Find the product of these whole numbers: 23 x 54.
9. Round to the nearest whole number: 86.256
10. Write a word name for the number in the sentence: In a recent year, the average salary of a player in the NBA was $1,867,000.
11. Kyle has to be at school by 8:00 a.m. He drives 30 mph and lives 5 miles from school. a. How long will it take him to get to school?
b. What time must Kyle leave for school to arrive five minutes early?
12. Dr. Schindler orders 20 mg. of Paxil in the morning and 20 mg. at night. He wants his patient to be on this medication for 30 days. How many 20 mg. pills will the patient receive from the pharmacist?
13. Amie needs to buy a graphing calculator for college. The one she wants costs $110. Amie only has $65. If she budgets $15 from her paycheck each week, how long will Amie need to save before she can buy the calculator?
Revised 2016 Sharon Stiehm Whole Number Problems - Answer Sheet
1. False. We use whole numbers to count objects. It is impossible to count objects using negative numbers. Therefore, negative numbers are not included and zero is the smallest whole number.
2. True. Whole numbers increase to infinity.
3. 6 (2356)
4. 5 (2356)
5. Answers will vary; however, the number 250,000,000 is written as two-hundred- fifty million.
6. Connor needs to gain 20 pounds and he has 4 months to do it. Divide the number of pounds by the number of months to find how many pounds he needs to gain each month. 20 ÷ 4 = 5. Answer: Connor will need to gain five pounds per month.
7. 702 123 556 2 + 21 702
8. 1,242 23 92 x 54 + 1150 92 1242
9. 86 To round to the nearest tenth, find the tenths place (86.256). Look at the number to the right. If the number is five or more, round up. If it is four or less, round down by dropping the decimal numbers.
10. One-million, eight-hundred sixty-seven thousand dollars.
11. a. Ten minutes. Use the distance formula (D = r t) Distance is equal to rate multiplied by the amount of time. If his distance is 5 miles and his rate is 30 mph, solve for “t” (time). (Express 30 mph as 30 ÷ 60 or 30/60) 30/60 t = 5 miles 1/2 t = 5 (Multiply both sides by 2 to eliminate the fraction) t = 10 Answer: It takes Kyle 10 minutes to get to school.
b. 7:45 A.M. If it takes Kyle ten minutes to get to school by 8:00, then he must leave by 7:45 to arrive five minutes early at 7:55.
Revised 2016 Sharon Stiehm 12. 60 20-mg pills. If the patient is taking two pills a day for 30 days, multiply the number of pills per day by the number of days.
13. Three weeks. Amie already has $65, subtract this amount from $110 and she only needs $45. If Amie budgets $15 per week, divide the amount needed by the amount budgeted each week, 45 ÷ 15 = 3. In three weeks, Amie will have enough money to buy the graphing calculator.
Revised 2016 Sharon Stiehm Reading Fractions
Fractions are a useful way to express remainders in math problems. The number .25 can also be written as 1/4. Five-tenths written in fraction form is 5/10 or when reduced to lowest terms is 1/2.
Reducing Fractions To reduce a fraction, follow these steps: 1. Find the “largest common factor” (LCF) for the numerator and denominator. 2. Then divide the numerator and denominator by this number. For example in the fraction 21/49, the LCF is 7. 21 ÷ 7 = 3 and 49 ÷7 = 7. Therefore, 21/49 reduced to lowest terms is 3/7.
Mixed Numbers In a mixed number, a whole number is written next to a proper fraction. Examples are 1½, 3¾, and 1¼. To change an improper fraction into a mixed number, follow these steps: 1. Divide the numerator by the denominator. The whole number in the quotient is the whole number portion of the mixed number. 2. Any remainder in the division problem is placed over the divisor and becomes the fraction portion of the mixed number.
Proper Fractions In a proper fraction, the numerator is smaller than the denominator. Examples are 1/3, 2/5, and 7/19.
Improper Fractions In an improper fraction, the numerator is greater than or equal to the denominator. Examples of improper fractions are 9/4, 12/3, 4/4, and 23/7. To change a mixed number into an improper fraction, follow these steps: 1. Multiply the whole number by the denominator. 2. Add the numerator to the result of step 1. 3. The answer from step 2 becomes the numerator. 4. The denominator remains the same.
Adding and Subtracting Fractions Fractions with the same denominator are simple to add and subtract. Follow these steps if the denominators are the same: 1. Add or subtract the numerators. 2. The denominator remains the same. 3. Reduce the final answer to lowest terms.
Revised 2016 Sharon Stiehm Fractions that do not have the same denominator must be changed to the least common denominator (LCD). To find the LCD of 2/3 + 7/24, follow these steps: 1. The LCD is the smallest number that is evenly divisible by both denominators. (The LCD for 3 and 24 is 24.) 2. The number you used to multiply the denominator of a fraction is the same number you used to multiply the numerator of that fraction. (Multiply the denominator of 3 by 8 to equal 24 then multiply the numerator of 2 by 8. The fraction 2/3 is changed to 16/24.) 3. Add or subtract the numerators. (16 + 7 = 23) 4. The denominator remains the same. (Answer: 23/24) 5. If necessary, reduce to lowest terms. (In this example 23/24 is in lowest terms.)
Multiplying Fractions To multiply a fraction, follow these steps: 1. Check to see if it is possible to cross-cancel. 2. Multiply the numerators. 3. Multiply the denominators. 4. Reduce to lowest terms.
Dividing Fractions
To divide fractions, follow these steps: 1. Use the reciprocal of the second fraction (interchange the placement of the numerator and the denominator). 2. Cross-cancel if possible. 3. Multiply the numerators. 4. Multiply the denominators. 5. Reduce to lowest terms.
Revised 2016 Sharon Stiehm Fraction Problems
1. Change 21/9 into a mixed number.
2. Reduce to lowest terms: 4/12.
3. Change the fraction to a whole number or reduce: 49/7.
4. Change the fraction to a whole number or reduce: 12/3.
5. What does 4/4 equal?
6. Change this mixed number into an improper fraction: 2¾.
7. Change this improper fraction into a mixed number: 14/3.
8. 1/2 x 3/4 =
9. 1/2 ÷ 3/4 =
10. 1¾ x 2/3 =
11. 1¾÷ 2/3 =
12. 4 x 1/3 =
13. 2/3 – 4/5 =
14. 12/42 + 2/56 =
15. 4 ÷ 1/2 =
16. (1/4 x 2/3) ÷ 1/2 =
17. (1/2 + 2/5)(4/5 + 1 2/3) =
18. (2/3 x 1/2) + 2/3 – 3/4 =
19. 1/2 – 2/3 + 3/2 =
20. (1 – 2/3)(5 + 2 2/3) =
Revised 2016 Sharon Stiehm Fractions – Answers
1. Divide 21 by 9. The remainder is the fraction. 2 3/9 or 2 1/3
2. 1/3 4. 4
3. 7 5. 1
6. Multiply 2 by the denominator of 4. Then add the numerator of 3 to get 11. This is numerator. The denominator remains 4. The
answer is 11/4.
7. Divide 14 by 3. The remainder of 2 is placed over the divisor of 3 as
a fraction. Answer: 4 2/3
8. Multiply numerators, then multiply denominators. Answer: 3/8
9. 4/6 or 2/3
10. First, change the mixed fraction to an improper fraction … 7/4 x 2/3.
Cross-cancel to get 7/2 x 1/3 = 7/6 or 1 1/6
11. First, change the mixed fraction to an improper fraction … 7/4 ÷ 2/3.
Take the reciprocal of the second fraction and multiply. 7/4 x 3/2 =
21/8 or 2 5/8
12. 4/1 x 1/3 = 4/3 or 1 1/3
Revised 2016 Sharon Stiehm 13. In adding or subtracting, fractions must have a common denominator. Multiply the numerator and denominator of 2/3 by 5. Multiply the numerator and denominator of 4/5 by 3. The equivalent fractions of 2/3 and 4/5, are now 10/15 and 12/15.
10/15 – 12/15 = - 2/15 (negative two-fifteenths).
14. 12/42 + 2/56 = 48/168 + 6/168 = 54/168 or 9/28
15. 4/1 ÷ ½ = 4/1 x 2/1 = 8
16. Reduce the fractions within the parenthesis; the problem now becomes (1/2 x 1/3) ÷ ½. Next, evaluate the fractions within the parenthesis to get (1/6) ÷ ½. Take the reciprocal of the second fraction, and problem becomes. 1/6 x 2/1. Reduce by cross-
canceling to get 1/3 x 1/1 = 1/3
17. (1/2 + 2/5) (4/5 + 1 2/3) = (5/10 + 4/10) (4/5 + 5/3) = (9/10) (12/15 +
25/15) = 9/10 x 37/15 = 3/10 x 37/5 = 111/50 or 2 11/50
18. (2/3 x ½) + 2/3 – ¾ = (1/3 x 1/1) + 2/3 – ¾ = 1/3 + 2/3 – ¾ = 3/3 – 9/12 = 12/12 –
9/12 = 3/12 = 1/4
19. ½ - 2/3 + 3/2 = 3/6 – 4/6 + 9/6 = 8/6 = 4/3 or 1 1/3
20. (1 – 2/3) (5 + 2 2/3) = (3/3 – 2/3) (15/3 + 8/3) = 1/3 x 23/3 = 23/9 or 2 5/9
Revised 2016 Sharon Stiehm More Fraction Problems
1. Multiply: 2/3 and 5/6
2. Divide: 13/5 by 4/5
3. Add: 1/2, 3/4, and 2/5
4. Subtract: 1/2 from 7/8
5. Write .5 as a fraction in lowest terms.
6. Write .75 as a fraction and reduce to lowest terms.
7. Evaluate: w/g if, w = 36 and g = 4.
8. Evaluate: a/b if a = 25 and b = 5.
9. Evaluate: 10 x (p/s) if p = 40 and s = 25.
10. Simplify: – 24/8
11. Add: - 3/4 and – 1/3
12. Evaluate: 5r + 6t – 4g – k if, r = 2; t = 1/2; g = 1/4; and k = 18.
13. Alex scored 1/2 of what Adam scored. If Adam received a 90%, what did Alex receive?
14. If Ranita weighs 3/4 of what Megan weighs, and Megan weighs 144 lbs., how much does Ranita weigh?
15. Find the fractional equivalent of .625 reduced to lowest terms.
16. Solve for x. x/6 = 5
17. Chris makes $1044 per month. If he wants to save 1/3 of his monthly earnings, how much will Chris have in eight months?
18. Find the average of these numbers: 2, 1, and 1/2.
19. Find the average of these numbers: 11/24, 1⅞, and 2/3.
Revised 2016 Sharon Stiehm More Fractions - Answer Sheet
1. Multiply the numerators then multiply the denominators. Then check to see if your final answer is reduced to lowest terms. 10/18 = 5/9
2. To divide fractions, use the reciprocal of the 2nd term then multiply. (First, check to see if the fractions can be reduced by cross-canceling.) 13/5 ÷ 4/5 = 13/5 x 5/4, Cross-cancel the denominator of five in the first fraction and the numerator of five in the second fraction. This leaves 13/1 times 1/4. Multiply the numerators then multiply the denominators. 13/1 x 1/4 = 13/4
3. Find the least common denominator (LCD). In this problem 1/2, 3/4, 2/5 the LCD is 20. Using the LCD 1/2 + 3/4 + 2/5 = 10/20 + 15/20 +8/20 = 33/20
4. The LCD is 8. Change 7/8 – 1/2 = 7/8 – 4/8 = 3/8
5. 5/10 = 1/2 7. 36/4 = 9
6. 75/100 = 3/4 8. 25/5 = 5
9. 10 x 40/25 = 10/1 x 8/5 = 16
10. – 24/8 = – 3
11. Find the LCD. In this problem the LCD is 12. – 9/12 – 4/12 = - 13/12
12. 5(2) + 6(1/2) – 4(1/4) – 18 = 10 + 3 – 1 – 18 = - 6
13. 1/2 x 90/100 (Cross cancel the denominator of 2 in the first fraction and the numerator of 90 in the second fraction to get 1/1 x 45/100). 1/1 x 45/100 = 45/100 = 45%
14. 3/4 x 144 = 3/4 x 144/1 After cross-canceling the problem is 3/1 x 36/1 = 108/1 = 108 Answer: Ranita weighs 108 lbs.
15. 0.625 is read as six-hundred twenty-five thousandths, which is 625/1000 in fraction form. 625/1000 is reduced to 5/8
16. Multiplying both sides by six (or for the left-side in fraction form, six over one) and cross- cancel. 6(x/6) = 6 (5); X = 30
17. 1/3 x $1044 = 348 per month. $348 x 8 = $2784
18. First change 2 and 1 into fraction form (2/1 and 1/1). Then find the LCD and add. In this problem the LCD is 2, 2/1 + 1/1 + 1/2 becomes 4/2 + 2/2 + 1/2. After adding the numerators, we have 7/2. To find the average of these numbers, divide 7/2 by 3 (the number of fractions in this problem). 7/2 ÷ 3 or 7/2 ÷ 3/1 = 7/2 x 1/3 = 7/6. The answer is 7/6
19. First change the mixed number of 1 7/8 to the improper fraction of 15/8. Then follow the same procedure as in number 18 by finding the LCD. In this problem the LCD is 24. 11/24 + 45/24 + 16/24 = 72/24 = 3/1 = 3
Revised 2016 Sharon Stiehm Decimal – Word Problems
Examples of using decimals in everyday life include… This Saturday everything in Kohl’s Department store is on sale for 20% (or 0.20) off the regular price. You can figure how much money you will save on an item that originally costs $25.00 by formulating a question. What is 20% of $25.00?
In a mathematical word problem, the word “is” means equals. To rewrite the example above as an equation, use a letter such as “X” to represent the unknown amount (this letter is called a variable) followed by an equal sign X =
Convert 20% to the decimal 0.20 X = 0.2…
The word “of” means multiply. Where you see the word “of” in the sentence is where you place a multiplication sign in the equation. (Multiplication signs include any of the following symbols x, ·, (amount to be multiplied). It is normal practice to use the dot or parenthesis when the equation has a variable X = 0.2 x 25; or X = 0.2 · 25; or X = 0.2 (25).
The word “and” is used for addition; however, the word “and” is also used in reading a decimal number (the decimal point is read as “and”). For example, 1.02 would be written as one and two-hundredths.
Decimal Name # of Places after the decimal Examples Tenths One Place .3 Hundredths Two Places .19 Thousandths Three Places .007 Ten-thousandths Four Places .0067 Hundred-thousandths Five Places .00183 Millionths Six Places .000023
1. What is one-tenth of 100? 2. Write the following number in word form: .0051 3. Write the following number in word form: 1.002 4. Of the following number, what digit is in the hundredth’s place? 135.2435 5. Of the following number, what digit is in the tenth’s place? 106.90? 6. Of the following number, what digit is in the thousandth’s place? 12. 245 7. What is one-hundredth of 2400? 8. Write the following number in word form: 10.2 9. Write two-thousand and four tenths in numerical form. 10. Write one-hundred and thirty-four hundredths in numerical form.
Answers: 1) 100 x .1 = 10; 2) Fifty-one, ten-thousandths; 3) One and two-thousandths; 4) 4; 5) 9; 6) 5; 7) .01 x 2400 = 24; 8) Ten and two-tenths; 9) 2,000.4; 10) 100.34
Revised - 2016 Sharon Stiehm Decimal Operations
1. In the number 34.683, which digit is in the tenths place?
2. Using the number above, which digit is in the tens place?
3. Add: 2.5 + 3.746 + .004 + 12.4 =
4. Subtract: 1277 – 82.78.
5. Write out the word name for 5.462
6. Divide: 54.7 by 2.5
7. Multiply: 4.00 x 68.125
8. Write three-fourths in decimal form.
9. What is 5699 divided by 10.25?
10. Perform the following operations: 3.75 – 2.4 - .001 – 23.6 =
11. Find the product of .001 and .35
12. Write the decimal notation for 2/5.
13. Normal body temperature is 98.6°. If Roberto’s temperature is 101.4°, how much is his temperature above normal?
14. A lab technician draws 9.85 mL of blood and uses 4.68 mL for lab testing. How much blood was discarded?
15. On a three-day trip, Jessica drove these distances: 110.35 miles the first day, 90.02 miles the second day, and 84.63 miles the third day. What was the average distance traveled per day?
16. If Sean fills his car with gas that is 105 cents a gallon, how many gallons will he put in his car if he only has $15.75? (Hint: use decimal form of 105 cents.)
17. A group of four students went out for lunch. They decided to split the bill equally four ways. The total cost for the meal was $48.00. If the students left a 15% tip, what did each student pay for lunch?
Revised - 2016 Sharon Stiehm Decimal Operations - Answer Sheet
1. 6
2. 3
3. Line up the decimals before adding. 2.500 3.746 .004 +12.400 18.650 or 18.65 4. 1194.22
5. Five and four-hundred sixty-two thousandths
6. 21.88
7. 272.5
8. Multiply the denominator by a number to equal a multiple of ten (10, 100, etc.). In this case, we multiply the denominator of 4 by 25 to equal 100. Then multiply the numerator by 25. (3/4 x 25/25 = 75/100 = .75)
9. 556
10. Line up the decimals before adding or subtracting.
3.75 1.350 1.349 - 2.40 - .001 -23.600 1.35 1.349 - 22.251
11. 0.00035
12. 0.4
13. 2.8º higher
14. 5.17mL of blood was discarded 15. 95 miles per day
16. 15 gallons 17. $13.80
Revised - 2016 Sharon Stiehm Decimal Problems
Numbers that are written as decimals such as 5.75 actually mean five and seventy-five one-hundredths. When division problems do not come out evenly, you can easily write the answer in a decimal from. Example: 15 ÷ 7 = 2.14286
1. What is 5.67 + 23.78?
2. Multiply: 45.7 by .005
3. Divide: 45.7 by .005
4. What is 34.6 – 23.76?
5. If Joe weighs 167.94 pounds, how many pounds does he need to gain before he weighs 170?
6. What is 24 divided by 16?
7. Francis wants to pay $50.00 on her phone bill for right now. If the total cost is $126.73. What is the remaining balance on her phone bill?
8. What is 23.5 + 23.5 – 23.5 + (– 23.5)?
9. John needs twenty-three units of UltraLente insulin before he goes to bed. If there is a total of 100 units in the bottle, how many units will be left?
10. Kwik Trip charges $.75 for a money order. If Jose buys three money orders. The amount of each money order is as follows: $12.37, $78.86, $153.74. How much will the money orders cost?
11. [4(7.5 – 4.9)] + 23.89 – 49 = ? (Remember when working with ( ), [ ], and { } begin with the innermost expression and work from the inside out.
12. Rachel goes to Wal-Mart to buy shampoo at $2.65 per bottle, toothpaste at $1.69 per tube, and a pair of sunglasses at $9.99 a pair. If she pays with a twenty- dollar bill, and there is no tax on these items, how much money will she receive back?
13. In the question 12, how much will Rachel receive if there is a 6 percent (.06) sales tax on all items purchased?
Revised - 2016 Sharon Stiehm 14. Sarah wants a Grilled Chicken Salad, fries, a large Coke at McDonald’s for lunch. The Grilled Chicken Salad Meal Deal costs $6.79. However, today there is a 15 percent discount on all individual menu items.
The individual prices for the items Sarah is ordering.
Grilled Chicken Salad $4.69
Fries $1.49
Large $1.59
Would it be less expensive for Sarah to purchase the Meal Deal or each item separately?
Revised - 2016 Sharon Stiehm Decimal - Answer Sheet
1. 5.67 + 23.78 29.45 (When adding or subtracting decimal numbers, always line up the decimal points vertically)
2. 45.7 x .005 .2285 (Multiply 457 by 5. From the right, count the number of decimal places in the first and second lines. This is the number of decimal places used in the product)
3. 45.7 ÷ .005 = 9140 (45.7 is the dividend and .005 is the divisor. The divisor must be a whole number. Move the decimal point of the divisor three places to the right. Then move the decimal point of the dividend three places to the right, adding two zeros after seven to complete the move. Complete the problem by dividing 45700 by 5 to obtain the quotient of 9140.)
4. 34.60 - 23.76 10.84
5. 170.00 - 167.94 2.06
6. 24 ÷ 16 = 1.5
7. $126.73 - 50.00 $ 76.73
8. 23.5 + 23.5 - 23.5 - 23.5 0
9. 100 - 23 77
10. $12.37 $.75 $244.97 78.86 x 3 + 2.25 +153.74 $2.25 $247.22 $244.97
Revised - 2016 Sharon Stiehm 11. [4(7.5 – 4.9)] + 23.89 – 49 = [4(2.6)] + 23.89 – 49 = (10.4) + 23.89 – 49 = 10.4 + 23.89 – 49 = 34.29 – 49 = – 14.71
12. 2.65 20.00 1.69 - 14.33 + 9.99 5.67 14.33
13. $14.33 x .06 = .8598 (round to the nearest cent, $0.86) $14.33 $20.00 + .86 - 15.19 $15.19 $ 4.81
14. $4.69 $7.77 $7.77 1.49 x .15 - 1.17 1.59 $1.1655 $6.60 $7.77 (round to the nearest cent)
Revised - 2016 Sharon Stiehm Percentage Problems
1. Twelve is what percent of 30?
2. Astronauts lose 1% of their bone mass for each month of weightlessness. If Matt is weightless in space for 5 months, how much bone mass did he lose?
3. What is 15% of 50?
4. Six is what percent of 20?
5. Change 0.45 into a percentage.
6. Change 2/3 into a percentage.
7. Change 5.1% into decimal form.
8. Change 5.1% into fractional form.
9. Change 1/8 into a percentage form.
10. 20% of what is 45?
11. If 30% of the students failed the exam and there are 180 students in the course, how many students passed the exam?
12. X = 85% (200-180) What does “x” equal?
13. If Katie weighs 125 lbs. and Marie weighs 10% more than Katie, how much does Marie weigh?
14. The skirt Mattie wants to buy is 25% off this week. If the original price was $49.99, what is the sale price?
15. Tony’s Bakery usually sells 250 desserts a day and typically 40 of these desserts are pies. What percent of the desserts sold will be pies?
16. Kurt buys 5 boxes of CDs in New Jersey where the sales tax rate is 7%. If each box of CD costs $14.95, how much tax will be charged? How much money will Kurt spend?
Revised - 2016 Sharon Stiehm Percentage Problems - Answers
1. 12 = 30x, 12 ÷ 30 = x, x = .4 or 40%
2. 5%
3. 7.5
4. 30%
5. 45%
6. 66.7%
7. 0.051
8. 51/1000
9. 12.5%
10. 225
11. 0.3 x 180 = 54, 180 – 54 = 126
12. x = 0.85 x 20, x = 17
13. 137.5 lbs. or rounded to 138 lbs.
14. $49.99 x .25 = $12.50, $49.99 - 12.50 = $37.49
15. 40 = 250x, 40 ÷ 250 = x, x = .16 or 16%
16. 14.95 x 5 = $74.75, $74.75 x .07 = 5.23 (tax), $74.75 + 5.23 = $79.98
Revised - 2016 Sharon Stiehm Arithmetic Problems
1. Is negative five (–5) a whole number?
2. Using whole numbers, complete the following problem: Kari weighs 120 lbs. and Rachel weighs 150 lbs. How many more pounds does Rachel weigh than Kari?
3. Simplify the following: 15/3 =
4. Multiply: 65 x 27 =
5. 12 + 235 + 4567 + 23,784 =
6. A bag of oranges weighs 27 lbs. A bag of apples weighs 32 lbs. Find the total weight of 16 bags of oranges and 5 bags of apples.
7. A rectangular lot measures 200 ft. x 600 ft. What is the area of the lot? What is the perimeter of the lot?
8. 482 + 1232 ÷ (50-42) = ____
9. Simplify the following: 5.9 + 10.5 = ___; 12.9 – 8.6= ___; 3.8 x 2.9 = ___
10. What is 5% of 20?
11. Jenny weighs 10% less than Josh, who weighs 165 pounds. How much does Jenny weigh?
12. A box contains 5,000 staples. How many staplers can you fill from the box if each stapler holds 250 staples?
13. James Dean was only 24 years old when he died. He was born in 1931. In what year did he die?
14. Two with an exponent of 5 is equal to what?
15. On Bailey’s first three math tests, she earned the following scores: 60%, 93%, and 86%. What is Bailey’s average score in math?
16. True or False, 154 is evenly divisible by (6) six.
Revised - 2016 Sharon Stiehm Arithmetic Problems - Answer Sheet
1. No, the smallest whole number is (0) zero.
2. Answer: 30 Subtract Kari’s weight from Rachel’s weight.
3. Answer: 15 ÷ 3 = 5
4. Answer: 1755 65 x 27 455 130 1755
5. Answer: 28,598 12 235 4,567 + 23,784 28,598
6. Answer: 592 First, multiply the number of bags by the number of pounds in each bag. Then, add the two numbers together.
7. Area is 120,000 square feet. (The formula for the area of a rectangle is length times width or A = wl). The perimeter is 1600 feet. (The formula for the perimeter of a rectangle is two times the length plus two times the width or 2L + 2w).
8. Answer: 636 First, do the subtraction within the parenthesis. Next, divide 1232 by the difference (the answer to the subtraction problem). Finally, add 482 to the quotient (the answer to the division problem).
9. 5.9 12.9 3.8 (Notice: the decimal points line up for +10.9 - 8.6 x 2.9 additional and subtraction problems) Answers: 16.4 4.3 11.02
10. In word problems, “of” means “to MULTIPLY” and “is” means “it EQUALS”. Change 5% to the decimal form of .05 and the equation becomes: X = .05 (20) or x = 1 (Note: Parentheses between numbers in the absence of an addition or subtraction symbol, means “to multiply”).
Revised - 2016 Sharon Stiehm 11. Answer: 148.5 lbs Ten percent of 165 pounds is 16.5 pounds. Subtract 16.5 from Josh’s weight to get Jenny’s weight.
12. Answer: Twenty staplers Divide 5,000 by 250. 5,000 ÷ 250 = 20
13. Answer: 1955 Add the year James was born to the age at which he died.
14. Answer: 32 (25 = 2 x 2 x 2 x 2 x 2 = 32)
15. 79.67% 60 93 + 86 239 ÷ 3 = 79.67%
16. False, if you divide 154 by 6, you get a number containing a decimal (154 ÷ 6 = 25.67). Therefore, one-hundred fifty-four (154) is not evenly divisible by six (6).
Revised - 2016 Sharon Stiehm