General Word Problems
Mistake 33
A certain car wash charges x dollars for the first y minutes of the wash, and $0.75 for each additional minute thereafter. If John washed his car for m minutes, how much did John pay for the car wash? 1) y - m = 0 2) x = 2.5 Because she is charged x dollars for the first y minutes, and not x dollars per y minutes, the cost of the car wash is the following x + 0.75(m-y) 1) y = m x + 0.75(m-m) x * 0.75(0) x Insufficient because we don't know x 2) x = 2.5
Consecutive integer word problems
Will always be +1 the number before. Sum of 3 consecutive integers: x + (x+1) + (x+2) = 3x +3
Moving from words to equations
is= equals was= equals has been= equals more= + years older = + years younger = - less = - times = x less than = - fewer = - as many = x factor = x Ex. George's age is three times Frank's age G= 3F
Mistake 32
Edward is home alone with a box full of cookies. When he wakes up, he eats 1/3 of the cookies plus 2 more. At lunch time, he eats 2/3 of the number of remaining cookies. At dinner, he eats the 4 remaining cookies. How many cookies were originally in the box? Solution: In the morning there are c cookies At the end of the morning there are c - (1/3c + 2) At the end of lunch time there are 2/3((c-(1/3c + 2)) At dinner he eats 4, and there are 0 remaining Therefore, we can create the following equation, that sums up all the cookies that have been eater (1/3c + 2) + 2/3((c-(1/3c + 2)) + 4 = C C = 24
Length problems
Establish variables for shorter piece and longer piece Lengths of individual elements must sum to a total length. Ex. 3 pieces x + y + z = total length Can be solved in much the same way as age problems
Mistake 4
Ex An electronics store began one week with x cell phones in stock .On Monday, the store sold 1/y of the cell phones and on Tuesday, it sold 1/z of the remaining phones Which of the following properly expresses the number of phones that the store had in stock at the end of the day on Tuesday? The company starts with x phones On monday it sells 1/y of the phones in stock. Therefore, by the end of Monday the remaining phones are: x - (1/y)x On tuesday at the end of the day, the remaining phones will be: Phones left by monday - 1/z* phones left by monday
Mistake 24
Ex: Brian works for a landscaping company and is paid x dollars for each lawn that he mows plus y cents commission for every second lawn that he mows. If Brian mows z lawns, where z is an even number > 1, which of the following represents the total number of dollars that he made? Mistake 1: "Brian makes x dollars per every lawn he moves". If he moves z lawns, he is paid xz dollars Mistake 2 "y cents commission for every second lawn that he moves". If he only makes money for every second lawn, it doesn't mean that he makes y cents after evert lawn he mows, it means that he is only compensated half of the time. Therefore, he gets z/2* y Mistake 3 Because y is in cents, and the questions ask for a dollar value, we will need to divide y by a 100. 5 cents is equivalent to 5/100 (0.05) dollars Final equation: xz + z/2(y/100) xz + zy/200
Compound interest problems
INterest paid on both original principal as well as accrued interest. FV = P(1 + r/n)^nt FV = future value P= principal r = interest rate n = number of compounding periods yearly t = time
Weight problems
Similar to age and length problems Of two dogs Gina and Tito, Gina weighs 20 pounds more than twice Tito's weight. If together they weigh 200 pounds, how much does Gina weigh? G= 2T + 20 G+T = 200 T= 200 - G G = 2(200-G) + 20 G = 140
Simple interest problems
Simple interest = principal x rate x time Make sure to match units.
Mistake 28
The product of the digits of a three-digit number x is 36. What is the sum of those three digits? a x b x c = 36 then we will have to play with the factors to come up with different numbers Factors are 1 ,2 x 2, 3 x 3 Possible numbers : 149, 194, 491, 661, 229... 1) x is an even number 2) x < 200 1) There are still many possibilities Ex: 914, 612,922... 2) x < 200 There are still many possibilities Ex: 166, 194, 149, 3) 166 and 194 meet both requirements
Mistake 17
When setting-up variables for problems where both the amounts and the prices are missing, use y and x as the amounts for the components, and the first letter of the components as the price Ex: A deli sells pints of two flavors of ice cream: vanilla and chocolate. If the deli purchases each pint of the two ice cream flavors for the same amount of money, did the store, last Monday, make a greater profit on the sale of the vanilla ice cream or the chocolate ice cream? sale price per pint of chocolate ice cream = c sale price per pint of vanilla ice cream = v number of pints sold of chocolate ice cream = x number of pints sold of vanilla ice cream = y
Splitting the cost
Usual an initial number of people plan to split the bill, but then some back out. The remaining people must pay a higher rate than planned. 10 people supposed to split (d/10), but only 8 did (d/8). What did the eight actually end up spending? (d/8 - d/10 = 5d/40 - 4d/40 = d/40 dollars
Word problems with inequalities
When we have an equation and an inequality each with the same variables, we can isolate one variable in terms of the other in the equation, and then substitute the isolated variable into the inequality.
Fractional parts of a whole must sum to the whole
When we remove some number of fractional parts from a whole, the sum of those fractional parts, plus what remains after they are removed, must equal that whole.
Consecutive multiples of integers
We can algebraically represent the consecutive integers of any number: For 5: x, (x+5), (x+10), (x+15), (x+20) If x is 25, it is true that the next consecutive integers will be 30 (x + 5), 35 (x + 10), 40 (x + 15), and 45 (x + 25) Remember that x is 25
Exponential growth problems
We can determine the amount of growth at the end of any growth period by calculating the product of the initial amount and the growth factor raised to an exponent that matches the growth period. For example, the amount of growth at the end of the 100th growth period is: (initial value) × (growth factor) to the power of 100. So, in general, to determine the total growth after a particular growth period, we can use the following formula: (initial value) × (growth factor) to the power of the growth period.
Mistake 16
Wet mixture problems are prone to change Ex: In an 80-quart mixture of heavy cream and water, fat represents 20 percent of the mixture. How many quarts of water must be added to the mixture to produce a mixture of only 10 percent fat? In the y axis: Mix 1, Water, Mix Final In the x axis: Percentage fat, amount of mixture (quarters), total fat
Dry mixture word problems
When breaking down a mixture problem, recognize the following attributes: a) The components of the mixture b) The units of each component c) The quantity of each component
Wet mixture word problems
When breaking down a wet mixture problem, recognize the following attributes: a) The components of the mixture b) The concentration of each component c) The quantity of each component
Mistake 6
When interest rate is an unknow variable, remember to express it as r/100 Ex: Carl invested $3,600 into a one-year investment paying simple annual interest. If, after 5 months, the investment has accrued $300 in interest, what is the interest rate on the investment? Principle * Interest Rate * Years = Interest earned 3600 * r/100 * 5/12 = 300 r = 20
Mistake 36
A cocktail consists of liquor and juice. The liquor costs $20 per liter, and the juice costs $1.75 per liter. If the cost to make one liter of the cocktail is $5.40, how many milliliters of liquor must be in a 250 ml serving of the cocktail? (1 liter = 1000 milliliters) Organize the data as in the picture x = number of liters of juice 1/4 - x = number of liters of liquor 1/4 - x is possible because we know that the total number of liters is 1/4 Following this approach we get a final equation: 5.4(0.25) = 20(0.25-x) x= 0.2 liters of juice Liters of liquor = Total liters - Liters of juice Liters of liquoe = 0.25 - 0.2 = 0.05 equivalent to 50ml My main mistake has been that I haven't been able to organize the data correctly. On the Y axis I should have included the components of the mixture (Liquor, Juice, Cocktail) On the X axis, I should have included the units of each component (Price per liter), and the quanity of each component (Liter), and finally, the total price
Mistake 31
A population of 10 ants grew by a certain factor x each day. What is the value of x? 1) Had the factor been halved, the ants would have grown to a population of 320 in 5 days 2) The average population growth for the first 5 days was2,046 ants per day 1) Under normal circumstances, at the end of day 5, the total number of ants would be 10x^5, but since x halves, at the end of day 5, ants will amount to 10(x/2)^2 2) If the average population growth for the first 5 days was 2046, then the total number of ants grown throughout the 5 days is 5 * 2046 = 10230 At the end of day 5 there were 10230 + 10 ants = 10240 Setting up an equation as in statement 1, 10x5^= 10240 x = 4
Mistake 12
Be 100% sure of the variables I am assigning and the equation I am building Ex: A certain internet cafe; charges $3.20 for the first 4 minutes of internet usage, and x dollars for every minute over 4 minutes. If Abby spent a total of 10 minutes on the internet, how much was she charged? 3.2 + x(10-4) 3.2 + 6x 3.2 dollars for the first 4 minutes x dollars for every minute after 10 represents the total number of minutes Abby spent on the cafe 1) The charge for each additional minute over the initial 4 minutes is 3 times the average per minute charge for the first 4 minutes x = 3(3.2/4) x = 2.4 3.2 + 6(2.4)= 17.6
Mistake 13
Be aware that when you set up equations and conduct multiple operations your are prone to make mistakes with the change of signs and with additions and substructions. Remember to keep focus throughout the whole problem
Mistake 8
Be careful with expressing the time that the investment accrues Ex: John deposited n dollars in an account that pays 2 percent annual interest, compounded semiannually. If the interest John earned at the end of 18 months is k dollars, what is n in terms of k? I = 0.02 N= 2 T = 18/12 If the interest John earned at the end of 18 months translates into 18/12 rather than into 18/24
Mistake 11
Be careful with objects in line questions because the positions of two objects can vary depending on how you look at the questions Ex: Two people, Adam and Ben, are waiting in a line to buy concert tickets. Adam is the 10th person counting from the beginning of the line, and Ben is the 15th person counting from the end of the line. If there are 5 people in between the two men, how many people could be waiting in the line? There are two ways to look at this: 1) If Adama is ahead of Ben, then 9 people --- A -- 5 people -- B --- 14 people => In total 30 people because A & B represent 1 person A is in the 10th position from the beginning and B is in the 15th position starting from the end 2) If Ben is ahead of Adam, 3 people --- B -- 5 people -- A -- 8 people => In total 18 people A is in the 10th position from the beginning and B is in the 15th position starting from the end
Mistake 3
Be careful with percentages, if I need to use the percentage given or if I need to use 1 - the percentage given Ex: Anthony is an employee of Godfather Airlines and although he must pay the full fare retail price when he spends $500 or less in a given year on tickets, his employee discount allows him to purchase tickets for 80 percent off the full fare retail price once he has spent $500. If Anthony received an 80% discount and paid $2500, it means that he paid just 20% of the actual price. Therefore: Actual price * 0.20 = 2500 Actual price = 2500 / 0.2
Mistake 2
Be very diligent at setting up the variables and remembering what they imply Ex: our colleagues planned to rent a car for a business trip and split the cost equally. If at the last minute, two of the colleagues do not attend the trip, the remaining people will each have to pay $40 more to rent the car In this questions I forgot to divide the total cost by the number of colleagues, which results in the cost per persom
Mistake 34
Carl is a collector of old quarters and nickels. If Carl were to add 12 quarters to his collection, he would have 3 times as many quarters as nickels. Does Carl currently have more quarters than nickels? 1) Carl has more than 6 nickel 2) Carl has fewer than 8 nickels The question is asking us if Q > N? From the prompt, we can deduce the following: Q + 12 = 3N Q = 3N - 12 If we substitute: 3N - 12 >N? 2N > 12? N > 6? If N is greater than 6 we have an answer 1) Sufficient 2) Not sufficient
Fraction word problems
Determine whether a fraction is being added to or removed from the original whole, or from some diminished portion of the original whole that remains. If we begin with x amount of anything and subtract 1/y of that amount, x - 1/y then x(y-1)/y will remain.
Mistake 18
Extract the most possible value out of the prompt Ex: If a positive three-digit number n is a multiple of 5, what is the value of n? From the prompt I know that the unit digit of n must be either 5 or 0. 1) The product of all the digits of n is 60 Integers that end in 5, or 0 and that abc=60 are 625, 345, 265 and 435 Not sufficient 2) The sum of the tens and units digits of n is 11 Since the units digit is either 5 or 0, then statement 2 allows us to calculate the following: a(100) + b(10) + c = n b+c = 11 If c = 0, then, b = 11, but since we can only have one digit, this result is not valid if c = 5, b = 6, this result is valid However, a can be any number from 1 to 9, so this statement is not sufficient Combining both statements, we know that the number must be either (625, 345, 265 435) and that the tenth digit must be 6. 265 is the answer
Age problems
First define present day age as a variable. Add to the variable if age is in the future. Subtract from the variable if age is in the past.
Mistake 5
For interest rate problems, in order to calculate the interest rate, use the following formula: Principle * Interest Rate * Years = Interest earned Ex: Tina invested $10 in an account that paid 5 percent simple interest each year. How much interest will Tina earn after 2 years? Principle * Interest Rate * Years = Interest earned 10 * 5/100 * 2 1
Mistake 27
For problems with vast amount of information, I need to set up a plan of action Ex: Jake set up a business at a lake conducting jet ski trips for visitors. On each trip, he carries one passenger, and each passenger pays him 10 dollars per mile. However, Jake has to pay $100 every 50 miles to refuel his jet ski. If Jake started the day with a full tank of gas and drove a total of 50 visitors an average of 5 km each that day, how much profit did Jake make? 1.6 km = 1 mile. This is a profit and loss problem, thus I need to calculate revenue and calculate costs Revenue can be calculated by multiplying the price per mile x the total miles navigates Price per km is 10 USD # KM navigated is 250/1.6 miles = 156.25 Revenue is 1562.5 USD Costs can be calculated by identifying when the Jack will need to refill his tank On miles 50, 100, and 150, he had to fill up the boat. Since each refill costs 100 USD, the total costs will be 300 US Profit = 1562.5 - 300 = 1262.5 My mistake was that I tried to calculate the profit per customer,
Mistake 21
If you can't build the equations, have another read at the prompt Example: At a certain ballpark are grandstand seats, 1/3 of the total seats are stadium-level seats, 1/6 are grandstand seats. How many of the seats in the stadium are stadium-level seats? 1) Define variables T = total # seats G = # grandstand seats S = # stadium seats 2) Define equations G = 1/3T S = 1/6T 1)) 1,200 grandstand seats were added to the stadium, grandstand seats would represent 1/4 of total seats G + 1200 = 1/4(T + 1200) I need to remember to add 1200 to T because by increasing G by 1200, T increases by 1200
Mistake 22
In data sufficiency questions, remember to have a look at the information you have from the prompt before confirming that one statement is not sufficient. Often you will need to plot the information from one statement into the general prompy Ex: At his favorite store, Tom bought a combination of boots, shoes, and sneakers, spending a total of $800. How much money did Tom spend on shoes? B + SN + SH = 800 1) The cost of the boots and sneakers represented 2/5 of the total amount Tom spent 2) The money Tom spent on shoes was $160 more than the money he spent on sneakers and boots combined. 1) B + SN = 2/5(800) B + SN = 320 SH + 320 = 800 SH = 480 I forgot to take the step in bold 2) SH = 160 + SN + B SN + B = 800 - SH SH = 160 + 800 - SH 2SH = 960 SH = 480
Mistake 10
In data sufficiency word problem questions where inequalities are involved, make sure to check the inequalities after every piece of information to see if the relationship is meet Ex: Penny and Nicole collect old pennies and nickels. Penny has more pennies than nickels. Nicole has more nickels than pennies. Who has more coins? Penny a = # pennies b = # nickels a > b Nicole c = # pennies d = # nickels d > c The questions is really asking is a + b > c + d 1) The number of pennies Penny has is 1 more than twice the number of nickels Nicole has a = 1 + 2d We can plot in a into the the general questions 1 + 2d + b > c + d 1 + d + b > c By looking at the prompt we know that d > c, therefore, we can confirm that 1 + d + b > c 2) The number of pennies Nicole has is 1 more than twice the number of nickels Penny has.
Objects in a line
In general, if you (or anyone) are the mth person counted from the beginning of the line and the nth person counted from the end of the line, then the number of people waiting in the line is m + n - 1 Ex: If you are the 10th, counting from the beginning of the line, there are 9 people in front of you. If you are the 15th person, counting from the end of the line, there are 14 people behind you. Therefore there are 24 people in the line. With the formula, 10 + 15 - 1 + 24
Mistake 19
In problems where there are constant rates, remember than you can set the constant rate as x, and adjust the equation accordingly Example: A particular type of grass grows by a constant amount every minute. The grass length was initially 2 yards. After 10 minutes, the grass was longer than it was after 6 minutes. How long was the grass after 12 minutes? After 1 minute the grass is 2 + x After 2 minutes the grass is 2 + 2x After 3 minutes the grass is 2 + 3x After 4 minutes the grass is 2 + 4x And after 12 minutes the grass is 2 + 12x Example: A gnome who was 16 inches tall swallowed a pill that caused him to grow by a certain factor x every minute for 6 minutes. After 6 minutes, the gnome was 1,024 inches tall. He then took another pill that reduced his height by a certain factor y every minute. If this reduced the gnome's height by twice as much as factor x increased his height, how many minutes did it take the gnome to grow to the maximum height and then shrink to a height less than his original height of 16 inches? Height at the end of minute 1: 16x Height at the end of minute 2: 16x(x)= 16x^2 Height at the end of minute 3: 16x2(x) = 16x^3
Mistake 1
Keep in mind the variable that I am being asked for so that I can optimize finding it over the other variables
Mistake 37
Kevin bought 15 scratch and win tickets for $7 each. Winning tickets pay three times the cost of the tickets and losing tickets pay nothing. After collecting the prize for his winning tickets, Kevin realized that he suffered a loss of $21 for the entire transaction. How many of Kevin’s 15 tickets were winning tickets This can be seen as a profit and loss problem Profit = Revenue - Expenses x: # winning tickets Because the money earned by the winning ticket is 3x the cost, and the cost per ticket is 7, the revenue per winning ticket is Revenue: 21x Because kevin bought 15 tickets at a cost of 7 each, he spent 105 in total Expenses: 105 Equation -21= 21x - 105 x = 84 / 21 = 4 4 is the number of winning tickets
Mistake 9
Make sure to substitute the variables we know into the inequalities given Ex: There are a total of 64 red and green marbles in a jar.If there are fewer than 44 green marbles in the jar, how many green marbles are in the jar? You can do: R + G = 64 R = 64 - G G= 64 - R G < 44 64 - R < 44 -R < - 20 R > 20 a) The number of green marbles in the jar is greater than 2 times the number of red marbles in the jar G > 2R You can do: R = 64 - G G > 2(64 - G) G > 128 - 2G 3G > 128 G > 42
Mistake 26
Make sure you answer the right question Example: Martha's car tire had a small hole punctured in it by a nail. Because of this hole, every pothole that she drove over caused the tire to lose 1/5 of its air. After she had driven over 4 potholes, what was the ratio of the air remaining in the tire to the air lost when she drove over the fourth pothole. Instead of using the air lost in the 4th pothole, I used the air lost in total, accounting for the air lost in the previous three potholes.
Mistake 15
Make sure you set up the whole equation before you conduct any operation Ex: In a profit problem, I forgot to account for the fact that the fixed costs occur every month rather than just once. The final equation should look something similar to the picture
Mistake 30
Martin owns a pizza shop, and his only monthly expenses are rent and utilities. If his rent is $4,000 per month, was Martin's profit greater than $1,500 in the month of June? 1) Martin's profit in June was 1/4th of its profits 2) Martin's utility expense in June was less than 1/3th of its profits Learnings: 1) Know very well where are your final equations are located in the whiteboard 2) When dealing with inequalities, try extreme cases to see if the inequality holds For instance, in this case, we obtain in the end that utilities are less than 1/12th of sales. By plotting in the extreme case of utilities accounting for 1/12th of sales in a different equation, we notice that profit will always be below 1500, which answers the question
Mistake 25
Plot in the variables you know into the inequalities to check for possible information Ex: A dairy sells each case of buttermilk for $90 and each case of ice cream for $20. If the dairy sold a total of 17 cases of buttermilk and ice cream last month, how many cases of buttermilk were sold last month? (Assume that fractions of a case cannot be sold.) Let b = the number of cases of buttermilk sold and c = the number of cases of ice cream sold b + c = 17 2) )Last month the dairy had more than $1,000 but less than $1,100 in revenue from buttermilk and ice cream sales The revenue from buttermilk and ice cream sales can be represented by 90b + 20c We can set up the following inequality: 1000< 90b + 20c < 1100 If we replace c by 17 - b i.e b= 17 - b we get 1000< 90b + 20(17 - b)< 1100 The result of the inequality is: 9.5 < b < 10.5 Therefore b must be 10 because it is the only integer
Price per item
Price per item x number of items = total cost of items purchased. (This is if all the items purchased were the same item and same price.)
Profit and loss problems
Profit = Total revenue - total cost Total cost = total fixed costs + total variable costs
Mistake 14
Remember that an answer can be obtained not only by having a final equation but also by using logic Ex: It takes Howard a minutes to read b pages; it takes Celia c minutes to read d pages. If abcd is different from 0, does Celia read at a faster rate than Howard a / b > c / d? 1) a is equal to c/4 a < c This means that the time that Howard reads is less than the time that Celia reads. Why? Because for C to be equal to A, C must be divided by 4 We can also let a = 10 and c/4 = 10., then c = 40 We don't have enough info to answer the question 2) b = (3/2)c b > c This means that the number of pages that Howards reads is greater than the number of pages that Celia reads. We don't have enough info to answer the question 3) If Howard reads more pages, using less time, then the Howard is a faster reader than Celia
Mistake 7
Remember that if I conduct a change in an equation, the change needs to apply across the entire equation Ex: sqr(x) / 100 -1 = y/200 I want to get rid of the denominators. Therefore, I multiply by the LCM (200) The result is 2 * sqr(x) - 200 = y
Money problems
Remember to multiply by 10 to clear decimals, but then divide by 10 again to go back to original decimal notation.
Useful matrix for mixture problems
See picture
Mistake 20
Solve for: A gnome who was 16 inches tall swallowed a pill that caused him to grow by a certain factor x every minute for 6 minutes. After 6 minutes, the gnome was 1,024 inches tall. He then took another pill that reduced his height by a certain factor y every minute. If this reduced the gnome's height by twice as much as factor x increased his height, how many minutes did it take the gnome to grow to the maximum height and then shrink to a height less than his original height of 16 inches? Solution: Increase The gnome is growing at constant factor. Therefore Height at the end of minute 1: 16x Height at the end of minute 2: 16x(x)= 16x^2 Height at the end of minute 3: 16x2(x) = 16x^3 At the end of the sixth minute, 16x^6 16x^6 = 1024 Square sixth of 1024 is 2 Therefore, the gnome doubles sizes every minute Since we know factor x is 2, that means that factor y is 4. Since y = 4 is the reduction factor, we have to divide the heights by 4, or multiply them by 1/4 Decrease Height at end of minute 7 = 1024/4=256 Height at end of minute 8 = 256/4=64 Height at end of minute 9 = 64/4=16 Height at end of minute 10 = 16/4=4 After 10 minutes, the gnome will be shorter than its original size
Mistake 23
Sometimes the equations that you need to construct are not very intuitive. If you feel you are missing something, there must be a different way of solving a problem Ex: A certain kickball team named The Good Guys; won 1/3 of their first 24 games. If the Good Guys lose no more than of their remaining games, what is the least number of games they must play to ensure they win more games than they lose? They won 1/3 of the games they played, so they won 8 and lost 16. We don't know the numbers of games left but we can represent that number with a variable R. If they will lose 1/9th of their remaining games, then future games lost is 1/9 * R If they will win 8/9th of their remaining games, then future games won is 8/9 * R The questions is asking what is the smallest number of R so that games won is higher than games lost. We can represent this with the following inequality: 8 + 8/9 * R > 16 + 1/9 * R 9(8 + 8/9 * R > 16 + 1/9 * R) 8R + 72 > 144 + R R > 10.5 The team must play a minimum of 11 games
Basic word problems involving only variables
Stick to what you know and treat variables like numbers. Common types of problems have total price of a mix of products or services, and have to determine price of an individual product or service.
Consecutive integer problems
The difference between two consecutive odd integers, or the difference between the two consecutive even integers is always 2 Odd: 13,15,17,19... Even: 2,4,6,8... Therefore, if we know the value sum of the sum of a given set of integers, and we know that all the integers within the sum are consecutive, we can obtain every given integer Ex: Three consecutive odd integers sum to 18. What is the smallest of these integers? 18 = x + (x+2) + (x + 4) 18 = 3x + 6 12 = 3x x = 4
Which salary should I choose/how much will I make?
This is when you are looking at 2 alternative ways to be compensated (will often compare a fixed salary with a commission one)
Mistake 29
Tom is shooting arrows with his friends for the second time. According to the rules, if a player successfully hits the target up to 10 times, he is awarded a total of x points, where x is a positive integer. If he hits the target more than 10 times, he is awarded 2x points and x/10 points per hit. Tom hits the target y times. If Tom's final score is an integer, is y > 10? 1) y < 20 and x is neither a multiple of 2 nor a multiple of 5 2) The first time Tom went shooting, he hit the target 9 times and received 19 points. 1) Because the final score must be an integer, Tom's score must have hit the target either 10 or less, or more than 10 times, but a multiple of 10, since the additional points are just a tenth of every point. To illustrate If y is 8, the number of points will be x which is always an integer if y is 12, the number of points will be x + x/10 which won't be an integer Because we know that the y (the number of shots) is less than 20, the only possible scenario where the points are an integer value is if y is <=10. Statement 1 is sufficient
Digit problems
Two digit numbers can be defined in the following fashion: 54 = 5(10) + 4(1) Therefore any two digit number can be written as: Two digit number: 10a + b
