Algebra Tutorial-word Problems

Algebra Tutorial—WORD PROBLEMS Word problems are easy to solve if you follow these three steps : 1 Read the problem at least twice; underline the key words that strike you or appears tricky to you. 2 Translate the word statements into mathematical symbols and equations or inequalities;.---that is ,set up equations or inequalities. 3 Think for a moment—get the mathematical procedure and proceed to solve the problem. Among these steps, the second step –translating word statements into math language-- appears difficult. This tutorial with several examples will make this process easy---at Algebra-1 level. ---------------------------------------------------------------------------------------------------Now let us start with examples: 1 Sarah is elder to John by 4 years. The sum of their ages is 26. Find the ages of Sarah and John. Here the ‘unknowns’ are Sarah’s age and John’s age Let us use symbols ‘x’ and ‘y’ for these unknowns. X ---the age of Sarah Y --the age of John Let us set up the equations. The sum of their ages is 26. - X + Y = 26 Sarah is elder to John by 4 years.  X= Y + 4 Now we have two equations in two unknowns. Let us solve : Substitute for x in the first equation: Y + 4 + Y = 26 2Y + 4 = 26 2Y = 22 Y + 11 X =15 Sarah is 15 years old and John ,11 years. 2 James counts the number of coins he has. He has nickels and dimes. The total number of coins is 16. The total value of the coins is $1.20 How many nickels and dimes James has? Let the number of nickels be X Let the number of dimes be Y Then the total number of coins  X + Y = 16 Now let us take the value of the coins. Since a nickel has a value of 5 cents and dime ,a value of 10 cents, The ‘value equation” is : - 5X + 10 Y = 120 [$1.20 is equal to 120 cents.] Let us solve the two equations to find x and y. The first equation appears simple.We shall use that to substitute for Y. 1

Y = 16 – X Then the second equation becomes: 5X + 10 ( 16 –X) = 120 This is an equation in one variable –that is X. Expand the second term using ‘Distributive property”: 5X + 160 – 10X = 120 -5X +160 = 120 -5X + 160 -160 = 120 – 160 -5X = -40 5X = 40 X =8 Since X +Y =16, Y=8 James has 8 nickels and 8 dimes. Check: 5X + 10Y = 5x 8 + 10 x 8 = 120 3 A circus company charges $1 for each child and $3 for adults.The total number of tickets sold in a particular day was 242. The manager counted the total amount collected that day. The amount was $422. How many children and how many adults visited the circus that day? This is similar to the coin problem given earlier. Let number of children tickets sold be X. Let the number of adult tickets sold be Y Total number of tickets: X + Y = 242 Value equation: X + 3 Y = 422 Let us subtract the second equation from the first: - 2Y = - 180 - Y = -90 Y = 90 X + Y = 242 X = 152. Check: X + 3Y = 152 + 3x90 = 422. -----------------------------------------------------------------------------------------------------For the following problems, we use some relationships or formulas from geometry. To recall: Area of a rectangle : A = length x width Perimeter of a rectangle: P = 2 (length + width) ----------------------------------------------------------------------------------------------------4 Sarah pays at the rate of $2 for mowing a lawn of area 300 square feet. She has a large lawn of 80 feet length and 30 feet width .How much she has to pay for mowing her lawn once? This problem has two parts: First find the area of her lawn: A = length x width = 80 x 30 = 2400 sq feet. The labor rate for mowing the lawn = $2 per 300 sqfeet. Find the cost for mowing the lawn: Cost = area /rate Rate = 300 square feet for $2 or 300/2 Cost = 2400/ [300/2] = [2400/300] x 2 = 8 x 2 =16 $

2

5

John wants to build a fence around his garden. His garden has length 80 feet and width of 40 feet. The contractor charges $12 for 100 feet length of the fence. How much John has to pay to build the fence. This is similar to the previous problem. Instead of area, we use perimeter formula. This problem has again two parts. First find the perimeter of the garden: P = 2 (length + width) = 2 ( 80 + 40 ) = 2 x 120 = 240 feet. Next find the cost: Rate for fencing is $12 for 100 feet. Cost = [240/100] x 12 = 2.4 x 12 = $ 28.80 or $29 [nearly]

We add more geometry related problem here. The area of a circle A = pi x radius x radius Circumference of a circle C = 2 x pi x radius Diameter of a circle D = 2 x radius You have to remember the value of a great constant in the Universe ‘pi’for the rest of your life!. Pi has a value which is 3.14159…… There are mathematicians and computers to find the value of pi to hundreds of decimal places. For our calculations, use: Pi = 3.14 6 The college stadium has a circular track with radius of 250 feet. The cost of preparing the track is $25 per foot of length. Find the total cost for preparing the track. Step1: find the circumference of the track: C = 2xpi x250 = 500 x 3.14 = 1570 feet Step 2: Cost of preparation: rate= $25 per foot Cost = 1570 x 25 = $ 39250 ------------------------------------------------------------------------------------------------------A concert hall has a semi-circular dais with diameter of 80 feet and a rectangular audience area of length 100 feet and width 80 feet. Find the total area if it costs $40 to make wooden flooring for 100 square feet,. Find the total cost. ‘Semicircular ‘ means ‘ a half circle.” Dais area (semicircular) = (1/2) pi x r x r = (3.14 x 40 x 40 )/2 = 2512 square feet 7

3

Audience area (rectangle) = 100 x 80 = 8000 sq feet. Total area = 2512 + 8000 = 10512 sq feet. Cost of wooden flooring = (10512 /100) x 40 = 420480/100 = $4204.8 8 Bill wants to make a card-board box with lid with length 3 feet ,width 2 feet and height of 1 foot. How many square feet of card board he should buy to make the box.Include extra 10% for edges and flaps. Top and bottom area= 2 ( 3 x 2 ) = 12 sq ft Side [lateral]areas = 2 (3 x1) + 2 (2x1) = 10 sq ft. Total area = 10 + 12 = 22 sq ft Add 10% =2.2 sq ft Total area required = 22 + 2.2 = 24.2 sq ft. Volume problems Volume of a rectangular box or space V = length x width x height Volume of a cylinder = pi x r x r x h Where pi = 3.14 r- radius h- height Volume of a sphere = (4/3) pi x r x r x r. 9 Mike has to store cereal grains of volume 128 cubic inches. If the base length of the box is 8 in, base width 2 in, find the required height for the box. Volume = 8 x 2 xheight = 128 Height = 128 / 16 = 8 in. 10 Bill wants to make a cylindrical barrel to store olive oil.The volume of oil is 2400 cu.feet. If the barrel is made with a base diameter of 12 feet, find the height of the barrel. Volume = pi x r x r xh = 3.14 x 6 x 6 xh = 2400 113 x h = 2400 Height = 2400 /113 = 23 feet. 11. A water fountain has a hemispherical base to collect the water.The diameter of the base is 20 feet. Find its volume. [ A hemi-sphere is half of a sphere.Its volume = (1/2) (4/3) pi x r x r x r] Volume = ( 2/3) 3.14 x 10 x 10 x 10 = 2.09 x 1000 = 2090 cubic feet 11 A light house on a beach is built by an architect with two simple shapes in mind. The tower is a cylinder with diameter of 20 feet and height 80 feet. The lamp house on the top is a hemisphere with diameter same as the tower.

4

An engineer is called up to make an air-conditioner for the light house.He needs to find the total volume of the light –house. Can you find the volume for him? Volume of tower V-tower = pi x r x r xh = 3.14 x 10 x 10 x 80 = 25210 cu ft Volume of the lamp house = (2/3) x pi x r x r x r = 0.66 x 3.14 x 10 x 10 x 10 = 2200 cu ft [approximate) Total volume = 27410 cuft Rate problems Rate problems use a simple formua: Distance traveled d = r x t Where r is the rate ,in this case speed in miles per hour or km per hour or meters/second or similar units. [distance/time] There are other rate problems too. Volume flow rate is the rate of pouring a liquid In this case ,rate is = gallons per minute or similar units. An important rate is labor rate: how much work is done per hour. Let us see a few examples. {see my other article: “Algebra tutorial –rate problems” –also in Scribd.com } 12 John drives his car for 4 hours at 55 miles per hour. The car gives give 22 miles per gallon of gas. What is the distance he travels? How much he spends for gas if it costs $4.5 per gallon ? We are going to use three rates given in the problem. Step1 distance traveled ; d = speed x time = 55 x 4 = 220 miles. Step2 gasoline consumed: v = fuel rate x distance = (1 gallon/22) x 220 miles = 220 / 22 = 10 gallons Step 3 Cost of gasoline

C= (cost/gallon) x volume = 4.5 x 10 = $ 45 Keep track of the units used: first miles, then gallons and then dollars in the results. 13 John and Mike start from Sacromento and from San Jose at the same time towards each other. John drives his car at 50 mph [miles per hour]

5

while Mike drives at 60 mph. The distance from Sacromento to San Jose is 440 miles. When will they meet? At what distance from San Jose? Let t be the time when both meet .Since they start at the same time, the time traveled by both is the same t hours. John travels x miles. Mike drives y miles. Since they travel towards each other, X + Y = 440 Distance traveled by John = X = 50 x t Distance traveled by Mike = Y= 60 x t X + Y = 440 50 t + 60 t = 440 110 t = 440 t = 4 hours. Mike travels from San Jose.he has traveled = 60 x4 =240 miles. They meet each other after 4 hours, at a distance of 240 miles from San Jose. 14. John and Alex start from the same place in Los Angeles, but at different times. John starts at 6 AM and drives his motorbike at 50 mph. Alex starts at 9 Am, goes in the same route as John. To catch John, Alex drives his motorbike at a speed of 65 mph. When will Alex meet John and at what distance from the starting point.? First understand the problem. John starts first and travels for t hours. Alex travels for lesser time since he starts at 9 AM, three hours behind. So, Alex travels for ( t -3 ) hours. Distance traveled by John d = 50 x t miles. If Alex catches with him ,he travels the same distance d ;but he travels for shorter time but at higher speed. Distance traveled by Alex = d = 65 ( t – 3) Now we can equate the two equations: d = 50 t = 65 ( t -3) Or 50 t = 65 t – 195 [Using distributive property] 50 t – 65 t = 65t – 65 t – 195 - 15 t = -195 t = 195/15 = 13 hours. The distance traveled d = 50 x t = 50 x 13 = 650 miles. Aled travels for 10 hours at 65 mph : d =65x10 = 650 miles. 15.A tiger sees a deer at a distance of 500 meters. The deer is running away from the tiger at speed of 200 meters per minute. The tiger sprints towards the deer at the speed of 300 ,meters per minute. Tigers have a limitation; they can sprint only for 1.5 minutes and then have to stop to get the breath. Can the tiger catch the deer? Distance traveled by the deer= 200 x t Distance from the tiger to the deer = d = 500 + 200t

6

Distance the tiger moves in time t = x = 300 t If the tiger should catch the deer , d =x Or 500 + 200 t = 300 t Let us solve for t : 500 = 300 t – 200t 500 = 100t or t = 5minutes. Tiger cannot catch the deer, since it will go out of breath by 1.5 minutes. The deer is safe for now!. 16 John can do the painting job for a house in 6 days. Mike can do the same job in 5 days. In how many days they can do the job if they work together? Let us find the rate at which they work; John does 1/ 6 th of work in one day. Mike does 1/5th of work in one day. If they work together , their combined rate R = 1/6 + 1/5 = (5 + 6)/30 work in a day. To find the number of days the work will take: W = rate x time Take w =1 then 1 = (11/30 ) x time Time = 30/11 = 2.8 days or nearly 3 days. [It is wrong to take the average of the two values of 6 and 5 days, giving 5.5 days.] Linear Equations Linear equations represent straight lines in a graph. Suppose we write: Y = a + b X, Note that for X =0, Y = a ‘a” is called the intercept. It is the initial value of Y. As x increases from zero, y will keep increasing if b is positive. As x increases, Y will decrease if b is negative. “b” is therefore ,called ‘slope’ . If slope is positive ,it is like going uphill. If slope is negative , it is like going downhill. When we construct a linear equation, we look for these two things---intercept and slope. 16 Sarah buys boxes of shirts on-line using the computer. Each box of shirts costs $ 22. The shipping charge is $ 8 for each order. Set up the equation and find the total cost if she orders 4 boxes and if she orders 6 boxes. What is the effective cost for each box in these two cases.? Bill amount = C = shipping charge + cost for order quantity C = 8 + 22 x Q Where Q is the order quantity.

7

This is a linear equation of the form: Y = a + b X For order quantity of 4 boxes, Q = 4 C = 8 + 22 x4 = $96 For order quantity of 6 boxes Q = 6 C = 8 + 22 x6 = 140 The effective cost of each box EC= total cost /Q For order qty of 4 boxes , EC = 96/4 = 24$ per box For order qty of 6 boxes , EC = 140/6 = 23.33$ per box. You save some amount by ordering more quantity ,if the shipping cost does not change with order quantity. 17 A train station on a hill is at the height of 820 feet above sea level. The gradient track from that station up the hill is 200 feet for every kilometer run of the train. If the next station is reached after 8 kms run of the train,find the height of the next station above sea level. Slope = 200 [feet per km run] Intercept = 820 Let us set up the linear equation: height H = 820 + 200 x L Where L is the run of the train on the track. For L =8 km H = 820 + 200 x 8 = 820 + 1600 = 2420 feet above sea level. The next station is located at 2420 feet above sea-level. 18 John jumps out of a plane with parachute unfurling.At the point of jump,his speed is zero.Dropping down,his speed increase at the rate of 4ft/sec .John touches the ground after 32 seconds.What is his speed during touch-down. Initial speed at the point of jump is zero intercept = 0 Speed S = 4 x t where ti sthe time in seconds. After touch down, speed S = 4 x 32 = 248 ft/second.

19 Amanda is planning for her son’s birthday party.. The hall rent is $ 120. She plans to provide food for $ 7 for each guest. What would be her expenses if she invites 30 guest and if she invites 40 guests. First set up the equation.The hall rent is fixed and is the same whether 30 guests arrive or 40 guests arrive or no guest comes. So, the intercept for this equation a = 120. The rate of food expense is $ 7 for each guest. Therefore slope is 7 ,Considering the number of guests as increasing quantity. The linear equation : Expense E = 120 + 7 x N Where N is the number of guests. Solving, for 30 guests Expense = 120 + 7 x 30 = 120 + 210 = 330$ For 40 guests, Expense = 120 + 7 x 40 = 120 + 280 = 400 $ [ The effective expense for each guest is: for 30 guests invited : 330/30 = 11$

8

for 40 guests invited: 400.40 = $10.] 20 Julie sets up a bakery in her garage.The cost of equipment for her bakery is $200. She spends $ 2 for each cake {fluor,sugar,electric power cost and so on]. She hopes to sell 500 cakes at $3 to local stores in the first week.Will she make profit in the first week? Let us set up the cost equation : C = 200 + 2 x X Revenue equation : R= 3xX Profit /loss equation: Profit/Loss = Revenue – cost = 3 X – (200 + 2 X) = 3X – 200 -2X = X – 200 Note that she plans to make and sell 500 cakes.Putting X = 500, We get Profit = 500 – 200 = 300 $ profit If she sells only 150 cakes, Loss = 150 – 200 = -50$ The last equation gives either profit or loss straightaway.If it is positive, it means profit.If it is negative, it means loss. ----------------------------------------------------------------------------------------

9