QRB501
Find A if A + 3A - 2 = 14 First, combine the unknown-value addends. Note that A is the same as 1A, so A + 3A = (1+3)A = 4A
A + 3A - 2 = 14 4A - 2 = 14 +2 +2 4A = 16 4A/4 = 16/4 A = 4 Check: A + 3A - 2 = 14 4 + 3(4) - 2 = 14 4 + 12 - 2 = 14 16 - 2 = 14 14 = 14
Solve an equation with more than one operation . Isolate the unknown value: (a) Add or subtract as necessary first. (b) Multiply or divide as necessary second. 2. Identify the solution: The solution is the number on the side opposite the isolated unknown value. 3. Check the solution: In the original equation, replace the unknown-value letter with the solution and perform the indicated operations.
3N - 1 = 14 3N - 1 = 14 +1 +1 3N = 15 3N/3 = 15/3 N = 5 Check: 3(5) - 1 = 14 15 - 1 = 14 14 = 14
Example 3: Your company has announced that you will receive a 3.2% raise. If your current salary is $42,560, how much will your raise be? What you know: Current salary = $42, 560 Rate of change = 3.2% What you are looking for: amount of raise Solution: Amount of raise = percent of change X original amount = 3.2%($42,560) The raise will be $1,361.92
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How to Find an equivalent formula by rearranging the formula
1. Determine which variable of the formula is to be isolated (solved for). 2. Highlight or mentally locate all instances of the variable to be isolated. 3. Treat all other variables of the formula as you would treat a number in an equation, and per- form normal steps for solving an equation. 4. If the isolated variable is on the right side of the equation, interchange the sides so that it ap- pears on the left side.
Verify that two fractions form a proportion
1. Find the two cross products. 2. Compare the two cross products. 3. If the cross products are equal, the two fractions form a proportion. Do 4/12 and 6/18 form a proportion? 4(18) = 72 12(6) = 72 Cross products are equal. 72 = 72 Fractions form a proportion.
Solve equations using addition or subtraction.
1. Isolate the unknown value or variable: (a) If the equation contains the sum of an unknown value and a known value, then subtract the known value from both sides of the equation. (b) If the equation contains the difference of an unknown value and a known value, then add the known value to both sides of the equation. 2. Identify the solution: The solution is the number on the side opposite the isolated unknown-value letter. 3. Check the solution: In the original equation, replace the unknown-value letter with the solution; perform the indicated operations; and verify that both sides of the equation are the same number.
Solve a proportion 1. Find the cross products. 2. Isolate the unknown by undoing the multiplication.
3/8 = 21/N 3N = 8(21) 3N = 168 3N/3 = 168/3 N = 56
Portion
part of the base
Mixed Percents
percents with mixed numbers or mixed decimals.
Find the value of A if A - 5 = 8 (A number decreased by 5 is 8.) The difference and the number being subtracted, 5, are known. Add 5 to both sides. The solution is 13. Check: Replace A with the solution, 13, and see if both sides are equal.
A - 5 = 8 +5 +5 A = 13 Check A - 5 =8 13 - 5 = 8 8=8
Equation
A mathematical statement in which two quantities are equal
Evaluate a formula (define)
A process to substitute known values for appropriate letters of the formula and perform the indicated operations to find the unknown value.
Base
The original number or one entire quantity
Unit Price
The price of a specified amount of a product. The unit price of a product is used when comparing prices of a product available in different quantities. The formula for finding the unit price is U = P/N , where U is the unit price of a specified amount of a product, P is the total price of the product, and N is the number of specified units contained in the product. The specified unit can be identified in many ways. The unit could be any measuring unit such as pounds (lb) or ounces (oz) or the number of items such as an indi- vidual snack cake in a package of cakes.
Proportion
Two fractions or ratios that are equal.
Write the fraction as a pecent
a) 67/100 67/100 (100%/1) = 67% b) 1/4 1/4(100%/1) = 25% c) 3 1/2 3 1/2 (100%/1) = 7/2 (100%/1) = 350% d) 7/4 7/4(100%/1) = 175% e) 2/3 2/3(100%/1) = 200%/3 = 66 2/3%
New Amount
the ending amount after an amount has changed (increased or decreased).
Example: Identify the given and missing elements for each example
a) 20% of 75 is what number? 20% = r/percent 75 = b/total what number = p/part b) what percent of 50 is 30? What percent = r/percent 50 = b/total 30 = p/part c) eight is 10% of what number? eight = p/part 10% = r/percent what number = b/total Use the identifying key words for rate (percent or %),base (total, original, associated with the word of ), and portion (part, associated with the word is).
known or given value
the known amounts or numbers in an equation
Percent of change
the percent by which a beginning amount has changed (increased or decreased).
Rate
the rate of the portions to the base expressed as a percent
Unknown or variable
the unknown amount of amounts that are represented as letters in an equation
Isolate Variable
to solve a formula for a desired variable
How to find the amount of increase or decrease from the beginning and ending amounts.
1. To find the amount of increase (when new amount is larger than beginning amount): Amount of increase = new amount - beginning amount 2. To find the amount of decrease (when new amount is smaller than beginning amount): Amount of decrease = beginning amount - new amount.
How to use the percentage formula to solve percentage problems
1. Identify and classify the two known values and the one unknown value. 2. Choose the appropriate percentage formula for finding the unknown value. 3. Substitute the known values into the formula. For the rate, use the decimal or fractional equivalent of the percent. 4. Perform the calculation indicated by the formula. 5. Interpret the result. If finding the rate, convert decimal or fractional equivalents of the rate to a percent.
How to find the amount of change (increase or decrease) from a percent of change
1. Identify the original or beginning amount and the percent or rate of change. 2. Multiply the decimal or fractional equivalent of the rate of change times the original or beginning amount. Amount of change = percent of change * original amount
Identify the rate, base, and portion
1. Identify the rate. Rate is usually written as a percent, but it may be a decimal or fraction. 2. Identify the base. Base is the total amount, original amount, or entire amount. The base often follows the preposition of. 3. Identify the portion. Portion can refer to the part, partial amount, amount of increase or decrease, or amount of change. It is a portion of the base. The portion often follows a form of the verb is.
Write a number as its percent equivalent
1. Multiply the number by 1 in the form of 100% 2. The product has a % symbol Write 0.3 as a percent. 0.3 = 0.3(100%) = 030.% = 30%
How to evaluate a formula
1. Write the formula. 2. Rewrite the formula substituting known values for the letters of the formula. 3. Solve the equation for the unknown letter or perform the indicated operations, applying the order of operations. 4. Interpret the solution within the context of the formula.
Write a percent as a number
1. divide the number by 1 in the form of 100% or multiply by 1/100% 2. The quotient does not have a % symbol
Example 3: In planning for a conference on Successful Small Business Practices, the organizers are anticipating that twice as many men as women will attend the conference. If they are expecting 600 to attend the conference, how many men and how many women are likely to attend? What you know: Total expected attendees: 600 Twice as many men as women are expected What you are looking for: Both the number of men and women are unknown. We can choose to represent the number of women expected to attend as W. Then, the number of men expected to attend is twice as many or 2W. Women: W Men: 2W Solution Plan: Men + Women = Total Attendees 2W + W = 600
2W + W = 600 3W = 600 3W/S = 600/3 W= 200 2W = 2(200) = 400 Check: Men + Women = 600 400 + 200 = 600 600 = 600 Conclusion: The organizers expect 400 men and 200 women to attend the conference
Find an equivalent formula by rearranging the formula
A formula can have as many variations as there are letters or variables in the formula. Using the techniques for solving equations, any missing number can be found no matter where it appears in the formula. Variations of formulas are desirable when the variation is used frequently. Also, in using an electronic spreadsheet, the missing number should be isolated on the left side of the equation. To isolate a variable is to solve for that variable.
Solve equations that are proportions
A proportion is based on two pairs of related quantities. The most common way to write proportions is to use fraction notation. A number written in fraction notation is also called a ratio. A ratio is the comparison of two numbers through division. When two fractions or ratios are equal, they form a proportion. An important property of proportions is that the cross products are equal. A cross product is the product of the numerator of one fraction times the denominator of another fraction of a proportion. In the proportion 1/2 = 2/4, one cross product is 1 X 4 and the other cross product is 2 X 2. Notice that the two cross products are both equal to 4. Let's look at other proportions.
Find the value of A if A/4 = 5 (A number divided by 4 is 5.) The quotient and divisor are known. The dividend is unknown. Multiply both sides of the equation by the divisor, 4. The solution is 20. Check: Replace A with the solution 20 and see if both sides are equal.
A/4 = 5 4(A/4) = 5(4) A = 20 Check: A/4 = 5 20/4 = 5 5=5
Solve the equation A/5 - 3 = 1 (A number divided by 5 and decreased by 3 is 1.) The equation contains both subtraction and division: Undo subtraction first, and then undo division.
A/5 - 3 = 1 +3 +3 A/5 = 4 5(A/5) = 4(5) A = 20 Check: A/5 - 3 = 1 20/5 - 3 = 1 4 - 3 = 1 1 = 1
Example 5: Stan sets aside 15% of his weekly income for rent. If he sets aside $150 each week, what is his weekly income? Identify the terms: The rate is the number written as a percent, 15%. The portion is given, $150; it is a portion of his weekly income, the unknown base.
B = P/R B = $150/15% B = 150/0.15 B = $1000 Stan's weekly income is $1,000 The rate is 15% and the portion is $150 (Figure 6-4). The base is the weekly income to be found. Convert 15% to a decimal equivalent. Divide.
Solve equations containing parentheses To solve an equation containing parentheses, we first write the equation in a form that contains no parentheses. 1. Eliminate the parentheses: (a) Multiply the number just outside the parentheses by each addend inside the parentheses. (b) Show the resulting products as addition or subtraction as indicated. 2. Solve the resulting equation.
Find A if 2(3A +1) = 14 2(3A +1) = 14 6A +2 = 14 6A + 2 = 14 6A + 2 = 14 -2 -2 6A = 12 6A/6 = 12/6 A = 2
Solve an equation when the unknown value occurs in two or more addends. 1. Combine the unknown-value addends when the addends are on the same side of the equal sign: (a) Add the numbers in each addend. (b) Represent the multiplication of their sum by the unknown value. 2. Solve the resulting equation.
Find A if 2A + 3A = 10 (2+3)A = 10 5A = 10 5A/5 = 10/5 A = 2
Evaluate a Formula
Formulas are procedures that have been used so frequently to solve certain types of prob- lems that they have become the accepted means of solving these problems. Formulas are composed of numbers, letters or variables that are used to represent unknown numbers, and operations that relate these known and unknown values. To evaluate a formula is to substi- tute known values for the appropriate letters of the formula and perform the indicated opera- tions to find the unknown value. Sometimes the equation must be solved to isolate the unknown value in the formula.
Portion can be called percentage
In a standard dictionary you will see percentage defined as "a fraction or ratio with 100 understood as the denominator," as "the result obtained by multiplying a quantity by a percent," and as "a portion or share in relation to a whole; a part." That is, the word percentage can refer both to the rate and the portion. This causes many to confuse the words percent and percentage. Because in written reports the word percentage is often used to identify the percent or rate instead of the portion, we will only use the word portion when referring to a part of the base.
What happens to the % (percent) sign?
In multiplying fractions we reduce or cancel common factors from a numerator to a denominator. Percent signs and other types of labels also cancel. %/% = 1
Tip: Arranging the proportion
Many business-related problems that involve pairs of numbers that are proportional are direct proportions. That means an increase in one amount causes an increase in the number that pairs with it. Or, a decrease in one amount causes a decrease in the second amount. In the preceding example, for 1 gallon of gas, the car can travel 23 miles. It is a direct pro- portion: More gas yields more miles. The pairs of values in a direct proportion can be arranged in other ways. Another way to arrange the pairs from the preceding example is across the equal sign. Each fraction will have the same units of measure. 1 gallon/16 gallon = 23 miles/M Miles 1 M = 16(23) M = 368 1 gallon & 23 Miles are pair 1 16 gallon & M Miles Pair 2
Solve the equation N + 15 = 25 (A number increased by 15 is 25.) The sum and one value are known. Subtract the known value, 15, from both sides. The solution is 10. Check: Replace N with the solution, 10, and see if both sides are equal.
N + 15 = 25 - 15 -15 N = 10 Check: N + 15 = 25 10+15 = 25 25=25
Example 1: Full-time employees at Charlie's Steakhouse work more hours per day than part-time employees. If the difference of working hours is 4 hours per day, and if part-timers work 6 hours per day, how many hours per day do full-timers work? What we know: Hours per day that part-timers work: 6 Difference between hours worked by full-timers and hours worked by part-timers: 4 What you are looking for: Hours per day that full-timers work: N Solution Plan: The word difference implies subtraction. Full-time hours - part-time hours = difference of hours N - 6 = 4
N - 6 = 4 +6 +6 N = 10 Check 10 - 6 =4 4 = 4 Conclusion The hours per day that full-timers work is 10.
Multiplying by 1 in the form of 100%
To write a number as its percent equivalent, identify the number as a fraction, whole number, or decimal. If the number is a whole number or decimal, multiply by 100% by using the shortcut rule for multiplying by 100. If the number is a fraction, multiply it by 1 in the form of 100%/1 In each case, the percent equivalent will be expressed with a percent symbol.
Solve
find the value of the unknown or variable that makes the equation true
Substraction
less than decreased by subtracted from difference between diminished by take away reduced by less/minus loss lower shrinks smaller than younger slower
Solve the equation 1. Isolate the unknown value or variable: (a) If the equation contains the product of the unknown factor and a known factor, then divide both sides of the equation by the known factor. (b) If the equation contains the quotient of the unknown value and the divisor, then multiply both sides of the equation by the divisor. 2. Identify the solution: The solution is the number on the side opposite the isolated unknown value. 3. Check the solution: In the original equation, replace the unknown- value letter with the solution; perform the indicated operations; and verify that both sides of the equation are the same number.
5N= 20 5N/5 = 20/5 N=4 5(4) = 20 20 = 20
A is the same as 1A When combining unknown-value addends, and one of the addends is A, it may help you to write A as 1A first.
A + 3A = 1A + 3A = 4A
Example 1: David Spear's salary increased from $58,240 to $63,190. What is the amount of increase?
Beginning amount = $58,240 New Amount = $63,190 Increase = new amount - beginning amount $63,190 - $58,240 = $4,950 Davids salary increase was $4,950
Example 2: A coat was marked down from $98 to $79. What is the amount of markdown?
Beginning amount = $98 New amount = $79 Decrease = beginning amount - new amount = $98 - $79 = $19 the coat was marked down $19
Equality
Equals is/was/are is equal to the result is what is left what remains the same as gives/giving leaves
Solve equations containing multiple unknown terms
In some equations, the unknown value may occur more than once. The simplest instance is when the unknown value occurs in two addends. We solve such equations by first combining these addends. Remember that , for instance, means 5 times A, or A+A+A+A+A To combine we add 2 and 3, to get 5, and then multiply 5 by A, to get Thus, is the same as 5A.
Tip: The process is important
Learn the process for solving applied problems with intuitive examples. In Examples 1 and 2 you may have been able to determine the solutions mentally, even intuitively. Learn the process for easier applications so that you can use the process for more complex applications. Many times a problem requires finding more than one unknown value. Our strategy will be to choose a letter to represent one unknown value. Using known facts, we can then express all other unknown values in terms of the one letter. For instance, if we know that twice as many men as women attended a conference, then we might represent the number of women as W and the number of men as 2W, twice as many as W.
Example 5: Your car gets 23 miles to a gallon of gas. How far can you go on 16 gallons of gas? What you know: Distance traveled using 1 gallon: 23 miles (pair 1) What you are looking for: Distance traveled using 16 gallons: M Miles (Pair 2) Solution Plan: Miles traveled per 16 gallons is proportional to miles traveled for each 1 gallon. 1 gallon/ 23 miles = 16 gallons/ M Miles Pair 1 Pair 2
Solution: 1/23 = 16/M 1 M = (16)(23) M = 368 Check: 1/23 = 16/368 (1)(368) = (23)(16) 368 = 368 Conclusion: You can travel 368 miles using 16 gallons of gas.
Variables
Letters used to represent unknown numbers
Solve Equation
Of the fractions 2/3 and 3/4, which one is proportional to 12/16? Find the cross products. Multiply. Not equal, not a proportion. Find the cross products. Multiply. Equal, proportional. Are 2/3 and 12/16 proportional? 2/3 = 12/16 2(16) = 3(12) 32 = 36 Are 3/4 and 12/16 proportional? 3/4 = 12/16 3(16) = 4(12) 48 + 48 3/4 is proportional to 12/16
Dealing with Parentheses in the order of operations and solving equations
Order of Operations To perform a series of calculations: 1. Perform the operations inside the parentheses or eliminate the parentheses by multiplying. 2. Perform multiplication and division as they appear from left to right. 3. Perform addition and subtraction as they appear from left to right. Solving Equations To solve an equation: 1. Eliminate parentheses by multiplying each addend inside the parentheses by the factor out- side the parentheses. 2. Undo addition or subtraction. 3. Undo multiplication or division. In the preceding example, examine the sequence of steps. To solve: Eliminate parentheses. Undo addition. Undo multiplication. To check: Add inside parentheses. Multiply.
Example 4: If 66 2/3% of the 900 employees in a company choose the Preferred Provider insurance plan, how many people from that company are enrolled in the plan? First, identify the terms. The rate is the percent, and the base is the total number of employees. The portion is the quantity of employees enrolled in the plan.
P = RB P = 66 2/3(900) P = 2/3(900/1) = 600
Example 3: During a special one-day sale, 600 customers bought the on-sale pizza. Of these customers, 20% used coupons. The manager will run the sale again the next day if more than 100 coupons were used. Should she run the sale again? What you know: total customers: 600 Coupon-using customers as a percent of total customers: 20% What you are looking for: Quantity of coupon- using customers. Should the manager run the sale again Solution plan: The quantity of coupon-Using customers is a portion of the base of total customers, atr a rate of 205 (figure 6-2) P=RB Quantity of coupon-using customers = RB Conclusion: The quantity of coupon-using is 120. Because 120 is more than 100, the manager should run the sale again
P = RB P = 20%(600) P = 0.2(600) P = 120 P is unknown; R = 20%; B = 600 Substitute known values. changes % to decimal equivalent multiply
Mixed Percents example
Percents can contain whole numbers, decimals, fractions, mixed numbers, or mixed decimals. Percents with mixed numbers and mixed decimals are often referred to as mixed percents. Examples are 33 1/3%, 0.05 3/4%, and 0.23 1/3%
Use the percentage formula to find the unknown value when two values are known
Portion = Rate X Base P = R(B) Base = Portion/Rate B = P/R Rate = Portion/Base R = P/B
Example 6: If 20 cars were sold from a lot that had 50 cars, what percent of the cars were sold?
R= P/B R = 20/50 R = 0.4 R = 0.4(100%) R = 40% Of the cars on the lot, 40% were sold The portion is 20; the base is 50 (Figure 6-5). The rate is the unknown to find. Divide. Convert to % equivalent.
Order of Operation Versus steps for solving equations
Recall that when two or more calculations are written symbolically, the operations are per- formed in a specified order. 1. Perform multiplication and division as they appear from left to right. 2. Perform addition and subtraction as they appear from left to right. To solve an equation, we undo the operations, so we work in reverse order. 1. Undo addition or subtraction. 2. Undo multiplication or division. In the example in the preceding How To box, examine the sequence of steps. To solve: Undo subtraction. To check: Multiply first. Undo multiplication. Subtract.
Example 2: Wanda plans to save 1/10 of her salary each week. If her weekly salary is $350, how much will she save each week? What you know: Salary = $350 Rate of saving: 1/10 What you are looking for: Amount to be saved: S Solution Plan: The word of implies multiplication. Amount to be saved = rate of saving X salary S = 1/10 ($350)
S = 1/10 ($350) S = $35 $35 = 1/10($350) $35 = $35 Conclusion: Wanda will save $35 per week
Example 3: Solve the formula S = C + M for C.
S = C + M S-M = C + M - M S-M = C C = S - M
Example 2: A DVD player that costs $85 sells for $129. What is the markup on the player? Use the formula S = C + M , where S is the selling price, C is the cost, and M is the markup. In some instances, the missing value is not the value that is isolated in the formula. After the known values are substituted into the formula, use the techniques for solving equations to find the missing value.
S = C + M $129 = $85 + M -85 -85 $44 = M The markup for the DVD player is $44
Example 1: Wal-Mart purchases a Sony plasma television for $875 and marks it up $400. What is the selling price of the television? Use the formula S = C + M , where S is the selling price, C is the cost, and M is the markup.
S = C + M S = $875 + $400 S = $1,275 The selling price for the television is $1,275
Example 6: The label on a container of concentrated weed killer gives directions to mix 3 ounces of weed killer with every 2 gallons of water. For 5 gallons of water, how many ounces of weed killer should you use? What you know: Amount of weed killer for 2 gallons of water: 3 ounces (Pair 1) What you are looking for: Amount of weed killer for 5 gallons of water: W ounces (Pair 2) Solution Plan: Amount of weed killer per 5 gallons is proportional to the amount of weed killer for each 2 gallons. 2 gallons/3 ounces = 5 gallons/ W ounces 2 gallons / 3 ounces (Pair 1) 5 gallons/W ounces (Pair 2)
Solution: 2/3 = 5/W 2W = (3)(5) 2W = 15 2W/2 = 15/2 W = 7 1/2 or (7.50) Check: 2/3 = 5/ 7 1/2 or (7.50) 2/3 = 5 / 15/2 2/3 = 5 (2/15) 2/3 = 2/3 Conclusion: You should use 7 1/2 ounces of weed killer for 5 gallons of water.
Example 4: Diane's Card Shop spent a total of $950 ordering 600 cards from Wit's End Co., whose humorous cards cost $1.75 each and whose nature cards cost $1.50 each. How many of each style of card did the card shop order? What you know: Total cost of cards: $950 Total numbers of cards: 600 Cost per humorous card: $1.75 Cost per nature card: $1.50 What you are looking for: There are two unknown facts, but we choose one - the number of humorous cards - to be represented by a letter, H. Number of humorous cards: H Knowing that the total number of cards is 600, we represent the number of nature cards as 600 minus the number of humorous cards, or Solution Plan Total cost = (cost per humorous card)(number of humorous cards) + (cost per nature card)(number of nature cards) 950 = (1.75)(H) + (1.50)(600 - H) 600 - H.
Solution: 950 = 1.75H + 1.50(600 - H) 950 = 1.75H + (1.50)(600) - 1.50H 950 = 1.75H + 900 - 1.50H 950 = 0.25H + 900 -900 -900 50 = 0.25H 50/0.25 = 0.25H/0.25 200 = H 600 - H = 600 - 200 = 400 Check: 950 = (1.75)(200) + (1.50)(600 - 200) 950 = (1.75)(2000 + (1.50)(400) 950 = 350 + 600 950 = 950 Conclusion: The card shop ordered 200 humorous cards and 400 nature cards. How to solve: Eliminate parentheses that show grouping. Multiply 1.50(600). Combine letter terms. Subtract 900 from both sides. Divide both sides by 0.25. The solution is 200, which represents the number of humorous cards. Subtract 200 from 600 to find 600 - H or 400, the number of nature cards. Substitute 200 in place of H. Then perform calculations using the order of operations. Subtract inside parentheses first.
Addition
The Sum of Plus/total increased by more/more then added to exceeds expands greater than gain/profit longer older heavier wilder taller
Ratio
The comparison of two numbers through division. Ratios are most often written as fractions.
Cross Product
The product of the numerator of one fraction times the denominator of the other fraction of a proportion.
Order of Operation
The specific order in which calculation must be performed to evaluate a series of calculations
Percent(s)
This relationship, called a percent, is used to solve many different types of business problems.The word percent means hundredths or out of 100 or per 100 or over 100 (in a fraction). That is, 44 percent means 44 hundredths, or 44 out of 100, or 44 per 100, or 44 over 100. We can write 44 hundredths as 0.44 or 44/100 The symbol for percent is %. You can write 44 percent using the percent symbol: 44%; using fractional notation: 44/100 or using decimal notation: 0.44.
Example 4: Find a variation of the formula U = P/N that is solved for P.
U = P/N N(U) = (P/N)N NU = P P = NU Isolate P. Multiply both sides of the equation by N. Simplify. N/N = 1. P(1) = P Interchange the sides of the equation. Formula variation
Did you know?
Using signed numbers you can combine the two previous rules into one rule for finding the amount of change. To find the amount of change: 1. Subtract the beginning amount from the new amount. 2. A positive result represents an increase. 3. A negative result represents a decrease. Look at Example 2 again. Amount of change = new amount - beginning amount =$79 - $98 = - $19 The negative result means that the change was a decrease.
We can relate the steps in our five-step problem-solving approach to writing and solving equations.
What You Know:Known or given facts What You Are Looking For: Unknown amounts (Assign a letter to represent an unknown amount. Other unknown amounts are written related to the assigned letter.) Solution Plan: Equation or relationship among the known and unknown facts Solution: Solving the equation Conclusion:Solution interpreted within the context of the problem
Formula
a procedure that has been used so frequently to solve certain types of problems that it has become the accepted means of solving the problems.
Formula 2
a relationship among quantities expressed in words or numbers and letters.
Percent
a standardized way of expressing quantities in relation to a standard unit of 100 (hundredth, per 100, out of 100, over 100).
More examples: write the decimal or whole number as a percent
a) 0.27 0.27 = 0.27(100%) = 027.% = 27% Multiply 0.27 by 100% (move the decimal point two places to the right). 0.27 as a percent is 27% b) 0.875 0.875 = 0.875(100%) = 087.5% = 87.5% Multiply 0.875 by 100% (move the decimal point two places to the right). 0.875 as a percent is 87.5% c) 1.73 1.73 = 1.73(100%) = 173.% = 173% Multiply 1.73 by 100% (move the decimal point two places to the right). 1.73 as a percent is 173% d) 0.004 0.004 = 0.004(100%) = 000.4% = 0.4% Multiply 0.004 by 100% (move the decimal point two places to the right). 0.004 as a percent is 0.4% e) 2 2 = 2(100%) = 200.% = 200% Multiply 2 by 100% (move the decimal point two places to the right). 2 as a percent is 200%
Examples: Solve the problems
a) 20% of 400 is what number? 20% = Rate 400 = Base Portion is unknown P = RB P = 0.2(400) P = 80 20% of 400 is 80 b) 20% of what number is 80? 20% = Rate 80 = Portion Base is unknown B=P/R B = 80/0.2 B = 400 20% of 400 is 80 c) 80 is what percent of 400? 80 = portion 400 = base Rate is unknown R = p/B R = 80/400 R = 0.2 or 20% 80 is 20% of 400 Very few percentage problems that you encounter in business tell you the values of P, R, and B directly. Percentage problems are usually written in words that must be interpreted before you can tell which form of the percentage formula you should use.
Examples: Write the percent as a decimal
a) 37% 37% / 100% = 0.37 Divide by 100 mentally. b) 26.5% 26.5% / 100% = 0.265 Divide by 100 mentally. c) 127% 127% / 100% = 1.27 Divide by 100 mentally. d) 7% 7% / 100% = 0.07 Divide by 100 mentally. e) 0.9% 0.9% / 100% = 0.009 Divide by 100 mentally. f) 2 19/20% 2.95% / 100% = 0.0295 Write the mixed number in front of the percent symbol as a mixed decimal before dividing by 100%. g) 167 1/3% 167.33% / 100% = 1.6733 or 1.673 rounded Write the mixed number in front of the percent symbol as a repeating decimal before dividing by 100.
division
divide(s) divided by divided into half of (divided by two) third of (divided by three) per
Isolate
perform systematic operations to both sides of the equation so that the unknown or variable is alone on one side of the equation. It's value is given on the other side of the equation.
Multiplication
times multiplied by of the product of twice (two times) double (two times) triple (two times) half of (1/2 times) third of (1/3 times)
