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Freezers made up what part of the appliances sold in January? If he wanted to finish the paper that What part of a day is 4 hours 20 minutes? A 1 Mrs.

The All Star Appliance Shop sold 10 refrigerators. What part of the membership is week? What part of the total job was done on Sunday?

A residential city block contains R one-family homes. Goldman finished the job by working Y hours on Sunday. What part of the buildings on this block is made up of one or two family houses?

Bach spent 9. Verbal Problems Involving Fractions 7. S two-family homes. A There was a total of 6 hours of 2 1 programming time. D The number of staff members is still C There are 30 pupils in the class. Since was done before breakfast. One yard is 36 inches. Divide by 2. Therefore This is the number of seniors.

This simplifies to A Change all measurements to inches. One gallon is 8 pints. Since 42 seniors voted for the Copacabana. D There were 40 students at the meeting. Since was spent on a hit 5 2 2 4 1 record. A After working for X hours. X hours are left out of a total of M hours. Multiply by 1.

They covered Multiply by 4. Divide by 3. Divide by 15 4. Divide by 7. Verbal Problems Involving Fractions Exercise 2 1. If attend the prom. Divide by 5.

A Since Multiply by 3. Since there are M men. They spent 5 1 4 4 of t or t the second week. C 36 minutes is 6. A There were 50 appliances sold in January. E Multiply by 7. Since Laurie already read for 8 hours. D Change all measurements to minutes. A 2 3 1 of or of the term paper was 3 4 2 1 completed on Saturday. Since 4 was completed on Friday. How many apples can be bought for c cents if n apples cost d cents? At how many revolutions per minute is the second gear turning?

A gear having 20 teeth turns at 30 revolutions per minute and is meshed with another gear having 25 teeth. Dehn drove miles during the first 5 months of the year. His younger brother weighs 50 pounds. How far on the other side of the fulcrum should he sit to balance the seesaw? How far apart in miles are two cities which are 5 4. A newspaper can be printed by m machines in h hours. If 2 of the machines are not working.

A 10 days B 12 days C 14 days D 16 days E 18 days 9. An army platoon has enough rations to last 20 men for 6 days. If a neighbor asks him to feed her dog as well. Alan has enough dog food to last his two dogs for three weeks. If 4 more men join the group. Variation 1. The numerator of the first fraction and the denominator of the second are called the extremes.

Observation often saves valuable time. Express the ratio of 1 hour to 1 day. Cross multiply. In solving a proportion.

There are 5 squares shaded out of 9. The ratio is 1. The denominator of the first fraction and the numerator of the second are called the means of the proportion. We refer to this as cross multiplying. Solve for x: The ratio of the shaded portion to unshaded portion is 5. A day contains 24 hours. In making this comparison. Find the ratio of the shaded portion to the unshaded portion.

Find the ratio of 1 ft. A team won 25 games in a 40 game season. Find the ratio of games won to games lost. The number of pounds of apples you download varies directly as the amount of money you spend. The number of pounds of butter you use in a cookie recipe varies directly as the number of cups of sugar you use. We must be very careful to compare quantities in the same order and in terms of the same units in both fractions.

This is direct. As the first decreases. As the first increases. This method often means a proportion can be solved at sight with no written computation at all.

A shortcut in the above example would be to observe what change takes place in the denominator and apply the same change to the numerator.

The more milk you download. If we compare miles with hours in the first fraction.

The more boys. Variation 2. We are comparing the number of bottles with cost. We are comparing the number of boys with the number of newspapers. You must always be sure that as one quantity increases or decreases. The denominator of the left fraction was multiplied by 4 to give the denominator of the right fraction.

Therefore we multiply the numerator by 4 as well to maintain the equality. Whenever two quantities vary directly. If b boys can deliver n newspapers in one hour. Find the cost. How many miles 2 1 4. What is the toll for a trip of miles on this road? If r planes can carry p passengers. Variation 3. The more people working. The product of the number of teeth and the revolutions per minute remains constant. E How long food.

Whenever two quantities vary inversely. Instead of dividing one quantity by the other and setting their quotients equal as we did in direct variation. A The number of teeth in a meshed gear varies inversely as the number of revolutions it makes per minute.

The fewer people working. The more people. The lighter the object. B The distance a weight is placed from the fulcrum of a balanced lever varies inversely as its weight.

The fewer men. The product of the diameter of a pulley and the number of revolutions per minute remains constant. The larger the diameter. The product of the number of people and the time worked remains constant.

C When two pulleys are connected by a belt. The product of the weight of the object and its distance from the fulcrum remains constant.

The heavier the object. D The number of people hired to work on a job varies inversely as the time needed to complete the job. The fewer people. The more teeth a gear has. The product of the number of people and the time it will last remains constant. The smaller the diameter.

The less teeth it has. This is inverse. If 3 men can paint a house in 2 days. There are several situations that are good examples of inverse variation. How many feet from the fulcrum must the heavier boy sit if the lighter boy is 8 feet from the fulcrum?

A 10 2 B C 10 3 9 D E 7 2 6 1 A gear with 20 teeth revolving at revolutions per minute is meshed with a second gear turning at revolutions per minute. If 10 of the children do not drink milk. Find the diameter. A field can be plowed by 8 machines in 6 hours. Two boys weighing 60 pounds and 80 pounds balance a seesaw. It is belted to a second pulley which revolves at revolutions per minute. If 3 machines are broken and cannot be used.

How many teeth does this gear have? A farmer has enough chicken feed to last 30 chickens for 4 days. If the quantities change in opposite directions. Variation In solving variation problems. If 10 more chickens are added. Exercise 4 Work out each problem. If only 10 oz. In the following exercises. A photograph negative measures 1 2 7 inches by 8 1 inches. The printed picture is to have its 2 longer dimension be 4 inches. How many pencils can be bought for D dollars if n pencils cost c cents?

How many miles apart are two cities that 1 are 3 inches apart on the map? How long should the shorter dimension be? How far from the fulcrum should a pound weight be placed in order to balance the lever?

If his salary continues at the same rate. If five more boys join in before the work begins. If the larger gear revolves at 20 revolutions per minute. Any fraction must be rounded up. A gear with 60 teeth is meshed to a gear with 40 teeth.

Variation 9. How many gallons of paint must be downloadd to paint a room containing square feet of wall space. B We compare miles to months. This is inverse variation. C The more men. C We compare inches to miles. D Number of machines times hours needed remains constant. D dollars is equivalent to D cents.

To avoid large numbers. Divide by p. Divide by a. D We compare gallons to miles. Variation Exercise 1 Exercise 2 1. Multiply by 2. C We compare cents to miles.

A The more cases. E Weight times distance from fulcrum remains constant. A Number of teeth times speed remains constant. B The more kilometers. B Number of machines times hours needed remains constant.

In p cases there will be 12p cans. Change 3 lb. A The more chickens. We compare cents with cans. E The less butter. A The more boys. Variation Retest 1. A Weight times distance from the fulcrum remains constant. D We compare pencils to dollars. C We compare gallons to square feet. B We compare inches to miles.

E Number of teeth times speed remains constant. The cost of n pencils is C 2 What percent of 40 is 16? Write as a fraction: What percent of 60 is 72? Percent 1. We multiply by This has the effect of putting the percent over and then simplifying the resulting fraction.

We multiply by and insert the percent sign. Learning the values in the following table will make your work with percent problems much easier. Then convert the decimal to a percent as explained on the previous page. Divide to two places only. Such fractions must be changed to decimals first by dividing the numerator by the denominator.

Percent Exercise 1 Work out each problem. Write 5 as an equivalent percent. When the fractional equivalent of the required percent is among those given in the previous chart.

It really pays to memorize those fractional equivalents. In finding a percent of a number. As we look at different types of percent problems. Percent Exercise 2 Work out each problem. Then it will become evident that we divide the given number by the given percent to solve.

Are you convinced that the common fraction equivalents in the previously 3 given chart should be memorized? It is often easier to translate the words of such a problem into an algebraic statement. Percent Exercise 3 Work out each problem. What percent of 72 is 16? If you have memorized the fractional equivalents 3 of common percents. It is often easier to put the part over the whole. A 5 20 10 25 15 B C What percent of 96 is 12?

What percent of y is x?

What percent of 48 is 48? Percent Exercise 4 Work out each problem. Let us look at one 2 example of each previously discussed problem. What percent of 90 is ? Percent Exercise 5 Work out each problem. A B C. What percent of 12 is 16? Write as a fraction in lowest terms: Multiply by 6.

E To change a decimal to a percent. A www. C What was the percent of decrease during this time? A A baseball team won 50 of the first 92 games played in a season. If the firm 2 originally employed workers. If the season consists of games. The basic sticker price on Mr. What percent of students graduating from Baker High will graduate from college? Study the examples on the following page carefully. What was his income that week? In Central City. At Baker High. How much must Mr.

What percent of the total price was made up of options? By what percent must it now increase its staff to return to the previous level? The population of Stormville has increased from What was the percent of increase?

There was an increase of The number of automobiles sold by the Cadcoln Dealership increased from one year to the following year. By what percent must it now decrease its sales staff to return to the usual number of salespersons? Find the percent of increase.

To increase by During the pre-holiday rush. Find the percent of decrease. Verbal Problems Involving Percent 1. The danger in this method is that the amount of discount. It is safer. What will be the net cost of a toaster. What did he make on the house? What was the marked price of the sweater?

How much did she forward to the cosmetics company? We must first find the total amount of her sales by asking: Silver sells shoes at the Emporium. This is called a commission. Audrey sells telephone order merchandise for a cosmetics company. Mitherz wishes to sell her home. How many papers did he deliver if they sell for 20 cents each?

What was his cost? This amount is called his profit. What is the sale price of the handbag? If a merchant sells an article for less than his cost. A loss is figured as a percent of his cost in the same manner we figured a profit in the previous examples. During a special sale.

At the last moment. What was the cost of the gown to the dress shop? What was the original price of the 3 ticket? Steve downloads a ticket to the opera. After using it for only a short time. How much money did the bike store give Alice? How much will Mrs. How much must Mrs. If Mrs. How many dollars tax does he pay? In Manorville. What will Mr. What was his total profit? At the Acme Cement Company. What are Mr. Find his commission.

What was the marked price? Rate of discount is figured on the original price. Verbal Problems Involving Percent 7. Find the other number. Find the average weight of these packages.

Valerie received test grades of 93 and 88 on her first two French tests. The average of W and another number is A. What grade must she get on the third test to have an average of 92? If Barbara drove for 4 hours at 50 miles per hour and then for 2 more hours at 60 miles per hour. Find the average of the first ten positive even integers. What is the average weekly salary paid to an employee? The average of any three consecutive multiples of 5 is the middle number.

Which of the following statements are always true? What grade must he earn on his fifth test in order to raise his average to 90? The average of any three consecutive odd integers is the middle integer. Mark has an average of 88 on his first four math tests.

The average of any three consecutive even integers is the middle integer. In finding averages. The 20th even integer is 40 use your fingers to count if needed and the 21st is If we are finding the average of an even number of terms. Averages 1. In this case.

To find the average of n numbers. Since these 40 addends are evenly spaced. Find the average of The above concept must be clearly understood as it would use up much too much time to add the 40 numbers and divide by Rewrite each number as a decimal before adding. To average fractions and decimals. Using the method described. Find the average of the first 40 positive even integers. The five men on a basketball team weigh Find the average of the first 5 positive integers that end in 3.

Find the average weight of these players. Find the average of a. Find the average of 3. Find the average of.

Find the missing number. The average of four numbers is The average of two numbers is 2x. What average must he have in his senior year to leave high school with an average of 92? On consecutive days. When the numbers are easy to work with. If three of the numbers are Averages 2. If the average of five consecutive integers is Exercise 2 Work out each problem. Therefore the missing number must be 12 units below Just watch your arithmetic.

What is the average weight of these girls? In driving from San Francisco to Los Angeles. What was his average rate. Exercise 3 Work out each problem. What was his average weekly wage for the summer? During the remaining six weeks of vacation. Find his average rate for the entire trip. Arthur drove for three hours at 60 miles per hour and for 4 hours at 55 miles per hour. During the first four weeks of summer vacation. In the Linwood School. Martin drove for 6 hours at an average rate of 50 miles per hour and for 2 hours at an average rate of 60 miles per hour.

Find the average yearly salary of these teachers. If M students each received a grade of P on a physics test and N students each received a grade of Q. In a certain gym class. The students of South High spent a day on the street collecting money to help cure birth defects. Find the average of the first 14 positive odd integers. Find the average amount contained in each of these cans. What is the average of 2x. What grade must he earn on the fourth test to have an average of 80 on these four tests?

The average of the first 4 positive integers that end in 2 is The average of the first twenty odd integers is Find the average of 4. The average of the first ten positive integers is 5.

In counting up the collections. Find the average height of these men. Represent the third number in terms of P. What was her average rate in miles per hour for the round trip? Karen drove 40 miles into the country at 40 miles per hour and returned home by bus at 20 miles per hour.

Since these are evenly spaced. So far. C The integers are 2. B These numbers are evenly spaced. E D The integers are 3. Mark is 8 points below 90 after the first four tests. A 93 is 1 above D Since 88 is 2 below E The average of any three numbers that are evenly spaced is the middle number. D 88 is 4 below A C 17 must be the middle integer. The numbers are The average of the first twenty positive integers is The first four positive integers that end in 2 are 2.

Averages Retest 1. C The integers are 1. Since the total distance traveled was 80 miles. C Karen drove for 1 hour into the country and returned home by bus in 2 hours. Their average is To add numbers with different signs.

If the signs are different. An even number of negative factors gives a positive product. Concepts of Algebra—Signed Numbers and Equations 1.

Add the following: Change the sign of the number to be subtracted and proceed with the rules for addition. Remember that subtracting is really adding the additive inverse.

To add numbers with the same sign. Subtract the following: If there is an odd number of negative factors. Divide the following: Find the following products: If the signs are the same. Be sure you know. What is the difference in their elevations? Find the average temperature for these hours. If the temperature rose 7 degrees during the next hour. Remove any parentheses by using the distributive law. Remember that whenever a term crosses the equal sign from one side of the equation to the other.

Determine the coefficient of the unknown by combining similar terms or factoring when terms cannot be combined. Divide both sides of the equation by the coefficient. Multiply by Collect all terms containing the unknown for which you are solving on the same side of the equal sign.

The techniques of solving an equation are not difficult. If there are fractions or decimals. Whether an equation involves numbers or only letters. Concepts of Algebra—Signed Numbers and Equations 2. Solve for a: This can be done by multiplying the top equation by 9 and the bottom equation by 5. Multiplying the first by 9. Since we wish to solve for x.

In doing this. If they had the same signs. The unknown can then be removed by adding or subtracting the two equations. Concepts of Algebra—Signed Numbers and Equations 3. If we were asked to solve for both x and y.

This can be done by multiplying one or both equations by suitable constants in order to make the coefficients of one of the unknowns the same.

Remember that multiplying all terms in an equation by the same constant does not change its value. When working with simultaneous equations. Multiply the first equation by d and the second by b to eliminate the y terms by addition.

The object is to eliminate one of the unknowns. Solve for y: At the level of this examination. From these two linear equations.

Remember that the difference of two perfect squares can always be factored. Factor out a common factor of x. Factor out a common factor of 2x. Concepts of Algebra—Signed Numbers and Equations 4. Since 8 is not a perfect square. If the middle term is also positive. We are now looking for two numbers that multiply to — Setting each factor equal to 0. To give —2 as a middle coefficient. In squaring each side of an equation. Then square both sides to eliminate the radical sign.

Solve the resulting equation. The entire side of the equation must be multiplied by itself. First get the radical alone on one side. Concepts of Algebra—Signed Numbers and Equations 5. Remember that all solutions to radical equations must be checked. B An odd number of negative signs gives a negative product. D An even number of negative signs gives a positive product. B Multiply by to clear decimals.

A Multiply by 8 to clear fractions. C B Multiply first equation by 3. D Multiply first equation by 3. A Multiply first equation by 3. D An odd number of negative signs gives a negative product. The sum of —4 and —5 is —9. A B C D E Express the number of miles covered by a train in one hour if it covers r miles in h hours.

If one book costs c dollars. If p pencils cost c cents. Represent the cost. How many dimes are there in n nickels and q quarters? Represent the price of a call lasting d minutes if d is more than 3.

The cost of a long-distance telephone call is c cents for the first three minutes and m cents for each additional minute. How many dollars must he pay if he uses the car for 5 days and drives miles? To the basic charge of c cents.

Kevin bought d dozen apples at c cents per apple and had 20 cents left. Since a foot contains 12 inches. Adding this to the 20 cents he has left. Find the total cost of sending a telegram of w words if the charge is c cents for the first 15 words and d cents for each additional word.

Just figure out what you would do if you had numbers and do exactly the same thing with the given letters. Therefore the number of cents in 2x — 1 dimes is 10 2x — 1 or 20x — Find the number of cents in 2x — 1 dimes. We must change everything to inches and add. To change dimes to cents we must multiply by Literal Expressions 1. In d dozen. If you understand the concepts of a problem in which numbers are given.

The computational processes are exactly the same. Express the number of inches in y yards. Since a yard contains 36 inches. Think that 7 dimes would be 7 times 10 or 70 cents. Represent the number of cents he had before this download. How many quarters are equivalent to n nickels and d dimes? Find his total earnings in a week in which he sells r dollars worth of merchandise. Find the charge. Express the number of days in w weeks and w days.

Represent the total cost. Assume that there is one teacher per class. E rc B The cost of mailing a package is c cents for the first b ounces and d cents for each additional ounce. Represent the total number of dollars paid to these men in a week. Represent the number of dollars he gets in a year. Represent the cost of one rose.

If a school consists of b boys. Represent the number of feet in y yards. How many dollars will Mr. Wilson pay if he used a boat from 3: The cost for developing and printing a roll of film is c cents for processing the roll and d cents for each print. If it takes T tablespoons of coffee to make c cups. How much will it cost. D This can be solved by a proportion. C The cost in cents of k pounds at c cents per pound is kc.

B The cost for the first 3 minutes is c cents. In q quarters. With m additional minutes. The charge. In h hours there are 60h minutes. To convert this to dollars. C In n nickels. E There are 60 minutes in an hour. C www. Principles and Procedures, 3e by Thomas F. Download Corto Maltese: Download Go Math! Grade 5 Teacher Edition Chapter 4: Denise M. Download Microservices in. NET Core: Download Obsessive Love: Download Stop Motion: Big Profits by John C. Download What Smart Students Know: Maximum Grades.

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