Ratios & Proportions:Lets nail this now
The Honey-Ginger candy bar has half the calories of the Almond-Chocolate candy bar, and has double the calories of a Peanut-Cherry candy bar. If one Honey-Ginger bar and one Almond-Chocolate bar and one Peanut-Cherry bar together contain a total of 840 kcal, what is the difference between the amount of calories in a Peanut-Cherry bar and in a Honey-Ginger bar? 120 140 240 260 280
280 Incorrect. The problem presents a ratio and a real number, so set up a ratio box: In order to easily find the ratio, plug in some easy numbers: If Honey-Ginger has 2 calories, then Almond-Chocolate has 4 calories, and Peanut-Cherry has 1 calorie. The actual calories are much greater than that (remember that the total of all 3 is 840 Kcalories), but the ratio remains the same: A : H : P is 4 : 2 : 1 The Real data of 840 Kcal total goes in the Total row. Fill the information in the box: ...................... Ratio ..Multiplier ..Concrete Ac ..................4.............. Hg.................. 2.............. Pc............ .......1.............. Total ...............7................................840 Now use the information in the box to find the multiplier and from there answer the question. ................. Ratio ..Multiplier ..Concrete Ac ...............4.............. Hg.............. 2............120......2*120=240 Pc...............1............120......1*120=120 Total ...........7............120...............840 The difference is 240-120 = 120kcal
In a photography exhibition, some photographs were taken by Octavia the photographer, and some photographs were framed by Jack the framer. Jack had framed 24 photographs taken by Octavia, and 12 photographs taken by other photographers. If 36 of the photographs in the exhibition were taken by Octavia, how many photographs were either framed by Jack or taken by Octavia? 36 48 72 96 108
36 Incorrect. This is a sets question. Since the question presents two sets with an interaction of A, B and both A and B, draw an "A/not A" 3×3 table. Place the information in the question in the table: Jack framed 24 of Octavia's photos - Goes in the Octavia / Jack box. Jack also framed 12 of other photographers photos - goes in Jack / Not Octavia box. 36 photographs were taken by Octavia - doesn't say whether these were framed by Jack or not, so it must include both. Therefore, this goes in the Octavia / total box. The table looks as following: .....................Octavia....Not Octavia...Total Jack...............24..............12 Not Jack....... Total..............36 Finally, what did the question ask? 48 Correct. Defining which boxes in the table contain the answer to the question requires some careful analysis: Photographs framed by Jack = Total Jack (top row = 24+12=36) Photographs taken by Octavia = Total Octavia = 36 It seems as if the question requires you to simply add the two: 36+36=72. The table looks as following: .....................Octavia....Not Octavia...Total Jack...............24..............12.................36 Not Jack....... Total..............36 However, notice an interesting fact - the 24 photographs which were both taken by Octavia and framed by Jack (those in the Octavia / Jack box) are actually counted twice: Once as part of the top row, and again as part of the left column. But in reality, these photographs should be counted only ONCE. Therefore, the correct calculation must discount them by subtracting them from the total: 36+36-24 = 48.
A secretary sends letters to the guest list of a certain dinner. In every letter there are three invitations. Every invitation allows a guest to bring along two friends. Assuming that all invited guests and their friends attend the dinner, how many letters should the secretary send so that 30 tables of 12 seats will be fully occupied? 20 40 60 80 120
40 Correct. This question is a proportions question. There is a fixed ratio between one letter and a certain number of people that arrive at the dinner due to that letter. First, make sure you determine this number correctly. Each invitation results in one invited guest + 2 friends - a total of three people. One letter results in three invited guests, or totaling 3+6=9 people. The proportion of letters to "occupied seats" at dinner, i.e., one to nine, must be equal to the proportion of x letters to 12×30 people at dinner. Solve for x. The fixed proportion between letters and people attending dinner is - letters/people = 1/9 = x/30*12 Where x is the required number of letters. Solve for x - x = 30*12/9 = 30*12/3*3 = 10*4/1
The ratio, by volume, of two pyramids is 28:64. If the volume of the bigger pyramid is 27,000 cubic meters larger than the volume of the other, what is the volume of the larger pyramid in cubic meters? 15,040 16,000 21,000 30,000 48,000
48,000 Correct. When the problem contains ratios and real numbers, use the ratio box to translate ratios to real numbers. Find out the loophole where you can get the multiplier. It may be the Total row, or, another combination of the rows. The real quantity in the question is the difference between the volumes of the pyramids. Add a row for the difference in the ratio box. Now, how are you going to fill the Ratio column? First, reduce the ratio 28:64 by a factor of 4 to 7:16. Then insert the 7 and the 16 in two rows. The ratio units for the difference is 16−7=9. It follows that the Multiplier is 3000, and the Real total is 16×3000=48,000. ...................... Ratio ..Multiplier .....Real Big......... ..........16...........*3000.....61*3000 ......................................................=48000 Small............... 7............. Difference.......9....27000/9=3000...27000 Total ................23................................36
A Metzhammer's bar contains only chocolate and ground almonds. The ratio, by weight, of chocolate to ground almonds is 4:5. What is the fraction of the ground almonds in a Metzhammer's bar? 4/9 5/9 4/5 5/4
5/9 Correct. This question asks about the fraction of almonds in the whole bar. A fraction represents part/whole . The fraction of almonds in the bar is part/whole = 5/5+4 = 5/9 .
In a tournament, Mitch plays 11 games. If Mitch wins a game he gets 4 yellow stars, and if he does not win he gets a fixed amount of red stars. At the end of the tournament, Mitch had 65 stars of both colors. If Mitch earned 33 more red stars than yellow stars, then how many games did Mitch win during the tournament? 4 5 6 7 8
A Correct. Numbers in the answer choices and a specific question ("...how many games...") call for Plugging In The Answers. You may feel like writing down one equation or more. This is just your algebraic urge, which is another stop sign for Reverse PI problems. Assume the amount in the answer choice is the number of games Mitch won (out of 11) and then follow the story in the problem. If everything fits - stop. Pick it. Otherwise - POE and move on, until you find an answer that works. Start with answer choice C. Assume that Mitch won 6 games. Then the total amount of stars he earned was 6x4 (yellow) plus (11-6)·N (red), where N is the number of red stars he got for every game not won. Therefore, 24+5·N=65. It follows that N is not an integer. POE C. Before you go on to Reverse PIug In, go a little further with this answer choice. N≈8 so that Mitch got ≈16 more red stars than yellow stars. Ask yourself whether the answer should be smaller or larger than 6. If you are not sure where to go, just pick a direction, and Reverse PIug In. If you chose to Reverse PIug In D, then 7×4+4·N=65⇒N≈9 and the difference between the number of red stars to the number of yellow stars is ≈6. The numbers get further away from the wanted goal, 33, indicating that D is in the wrong direction. Therefore POE D and E as well. The correct answer is A. If there are 4 games won, 4x4+7·N=65, and N=7. The difference between the number of red stars and yellow stars is 7x7-4x4=33, as required.
Most of the ratio GMAT problems consider concrete numbers, which are numbers that are associated with a particular thing, as well as ratios. That is, the information in the problem is about ratios while the question asks for concrete numbers, or vice versa. Consider the following problem: To make her biscotti more crispy, chef Suschard adds to the dough a mix of nuts containing walnuts, almonds, and pistachios at a ratio of 1 to 3 to 5. If chef Suschard adds a total of 36 ounces of mix to her dough, then how many ounces of almonds are in the mix? (A) 4 (B) 12 (C) 16 (D) 20 (E) 32
A Ratio by itself tells you nothing about concrete numbers. When the problem contains ratios and concrete numbers, use the ratio box to translate ratios to concrete numbers. Set up the ratio box like so: There are always three columns in the following order: Ratio, Multiplier, Concrete. Rows are the parts of the ratio that appear in the problem always followed by the Total row. In this case: Walnuts, Almonds, Pistachios, Total. --------- --- Ratio Multiplier Concrete Walnuts Almonds Pistachios Total Now fill in the information from the problem to the ratio box. Put ratios in the Ratio column and concrete numbers in the Concrete column. ...................... Ratio ..Multiplier ..Concrete Walnuts ..........1 Almonds......... 3 Pistachios .......5 Total .....................................................36 The Multiplier is the link between the ratio world and the concrete world. To figure it out you must have a number in the Ratio column and a corresponding number in the Concrete column. What is the Multiplier for the table above ? 1 2 3 4 5 Correct. Find out the loophole where you can get the multiplier. The Ratio column totals 1+3+5=9 at one end and 36 at the other end, in the Concrete column. So the multiplier must be ×4. ...................... Ratio ..Multiplier ..Concrete Walnuts ..........1..............*4 Almonds......... 3..............*4...............12 Pistachios .......5..............*4 Total ................9....................................36 The secret of the Ratio box is this: the multiplier is applicable for all parts of the ratio box. Once you have it you can get any piece of information the question asks about. For example, the concrete number of almonds in the mix is 3×4=12. The correct answer is B.
Sometimes you may need to set up an equation of two ratios, in order to solve the problem. An equation where a ratio equals a ratio is termed proportion. Take a look at the following problem; In one week, Coco the chimp eats four times his trainer's body weight in bananas. If Coco's trainer weighs 150 pounds, and if a banana weighs 8 ounces, then how many bananas did Coco eat last week ? (1 pound=16 ounces) (A) 600 (B) 800 (C) 1000 (D) 1200 (E) 2400 On the left side, form a fraction of pounds to bananas. 1 pound equals 16 ounces. Since each banana weighs 8 ounces, 1 pound equals two bananas, or a ratio of 1:2. On the right side, build the same fraction for Coco. Coco eats four times his trainer's body weight, hence 150·4=600 pounds of banana. Leave the number of bananas as x, as that's what the question asks for. The end result is the proportion: 1 lbs/2 bananas = 600 lbs/x bananas Now don't go bananas, just solve for x: 600 800 1000 1200 2400 Correct. Cross multiply to get x=600·2 = 1200 bananas. The answer is D.
A fixed ratio calls for setting up proportions. A proportion is an equation with two equal ratios, one on each side. Remember that a fraction is simply another way of writing a ratio. If the ratio of a:b is equal to the ratio of c:d, then setting a proportion between these two ratios will look like this: a/b = c/d Which of the following proportions do you find the most useful towards solving the question? Coco's weight to his trainer's weight Coco's trainer's body weight to the number of bananas pounds to number of bananas pounds to number of bananas You're right on track. Since the problem asks for the number of bananas, set up a proportion of pounds to number of bananas. Form two fractions, with pounds on top and number of bananas at the bottom. Although there may be more than one possible proportion that is relevant to solving this problem (i.e., ounces to pounds), the most important factor in working with proportions is to make sure the tops of the fractions have the same units, and so do the bottoms of the fractions.
Have you ever dreamed to be a bartender? Well in case you wondered, here's one recipe for a Margarita. "Mix 3 parts Tequila together with 2 parts orange liquor and 1 part fruit extract of your choice. Pour into a Margarita glass. Ice is optional. Enjoy" How many glasses of Margarita does this recipe make? 6 glasses As many as I like... Now answer the following question: What is the ratio of fruit extract to orange liquor in my Margarita? 1:3 1:2 3:2 2:1 3:1 Correct. A ratio is defined as part/part . In this case, 1/2.
As many as I like... The recipe gives you the relative amounts of ingredients, i.e., the ratio. It is not their real quantities. Therefore, you can make as many glasses of Margaritas as you like. All you have to do is adjust the quantities accordingly, such that their relative size is not hindered. Treat this ratio exactly as you would treat a fraction, reducing or expanding it as needed. In GMAT speak the ratio of Tequila to orange liquor to fruit extract is 3 to 2 to 1, or 3:2:1. Having said that, try to answer the following question. What part of my Margarita is Tequila? 3/6 2/3 3/2 2/1 3/1 3/6 Correct. A part or a fractional part is defined as (part/whole). In this case, 3/3+2+1 = 3/6 .
The atmosphere of a certain planet contains only three gases: Oxygen, Hydrogen and Nitrogen. What is the ratio of Oxygen to Hydrogen in the atmosphere of that planet? (1) Nitrogen makes up a fifth of the planet's atmosphere. (2) The ratio of Oxygen to Nitrogen in the planet's atmosphere is 3:2.
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked; Correct. This is a "what is the value of..." DS question. In this type of question, a statement will be sufficient only if it leads to a single value of the variable (or expression) you're asked about. The issue is combining ratios. Stat. (1): if Nitrogen makes up one fifth of the atmosphere, then Oxygen and Hydrogen make up the remaining four fifths. However, the internal ratio between Oxygen and Hydrogen is unknown, so Stat.(1)->IS->BCE. Stat. (2): without knowing the total ratio, or a connecting ratio with Hydrogen, it is impossible to find the ratio between Oxygen and Hydrogen. Stat.(2)->IS->CE. Stat. (1+2): write the two ratios in a diagram: ..............Oxygen : Hydrogen : Nitrogen 1st ratio ( ------ 4 --------- : 1 ) /·2 2nd ratio ( 3............................. : 2 ) The 1st ratio is a ratio of 4:1 between Hydrogen+Oxygen and Nitrogen. The 2nd ratio is one of 3:2 between Oxygen and Nitrogen. Combine the two ratios by equating the common member - in this case, Nitrogen. Multiply the 1st ratio times two, so the both ratios have "2" in the Nitrogen column. Oxygen : Hydrogen : Nitrogen 1st ratio ( ------ ....8 ------ : 2 ) 2nd ratio ( 3 ..........................: 2 ) Thus the ratio number for Oxygen+Hydrogen is 8, out of which 3 are Oxygen. Therefore, the ratio number of Hydrogen is 8-3=5, and the ratio of Oxygen to Hydrogen is 3:5. Stat.(1+2)->S->C.
Answer the following Non-GMAT question: Which of the following is true? I) 2.4/2.6 = 12/13 II) 0.25x/2y = x/8y III) 1+x/30 = 2+2x/15 I only II only I and II only II and III only I, II and III
C Correct. Expanding a fraction or reducing it does not change its value. To check whether two fractions are equal, try to expand/reduce one of the fractions to the form of the other. If you succeed, then the fractions are equal.
If p/3 = r/2 and r=0.75s, what is the ratio of p to r to s? 2:3:4 8:12:9 9:6:8 6:4:3 12:8:3
C Correct. This problem contains two ratios (given in the form of equations), and asks you to construct the threefold ratio. Instead of manipulating equations and accidentally reaching the wrong ration, plug in numbers that satisfy the equations. Since the problem asks for a ratio, which can be freely expanded and reduced, any p, r and s and will do, as long as the ratio between them is calculated accurately. Plug in good numbers for p and r. If p=3 (so that p/3 = 3/3 = 1 ), then r must be 2 (so that r/2 = 2/2 = 1 ). Now plug in r=2 into the last equation: 2 = 0.75s = 3/4s s=2*4/3 = 8/3 So the final ratio of p : r : s is 3 : 2 : 8/3. Expand the ratio times 3 to eliminate the fraction and get a result of 9 : 6 : 8.
In McDonald's farm, the ratio of the number of goats to the number of sheep is half the ratio of the number of cows to the number of goats in his farm. If the ratio of the number of goats to the number of sheep is 1:3, then what is the ratio of the number of sheep to the number of cows in the farm? 9:2 3:2 2:3 6:1 1:6
Correct. Break the problem into two steps: First, find the cows:goats ratio. The goats:sheep ratio (1:3) is half the cows:goats ratio, hence, the cows:goats ratio is double the goats:sheep ratio. This means that the ratio of cows:goats is 2:3. Next, write the two ratios in the following diagram to see what's going on: .................. Cows : Goats: Sheep 1st ratio .................... 1 ........: 3 2nd ratio .......( 2 .......: 3 ) In order to find the ratio of sheep to Cows, combine the two ratios: find a quantity that is common to the two (goats). Expand / reduce the ratios to equate the number representing the common quantity. Expand the first ratio by 3 to equate the common member - Goats. .................. Cows : Goats: Sheep : 1st ratio ..................( 3..... : 9 ) 2nd ratio .........( 2 :... 3 ) The ratio of sheep to cows is therefore 9:2.
What is the fraction of girls among the school children? (1) The ratio of boys to girls in school is 4:5. (2) The number of girls is 25% larger than the number of boys in school.
Correct. First, consider statement 1 alone. From this statement it follows that the fraction of girls is 5/(5+4). Therefore (1)->S->AD. Go on to consider statement 2 alone. This statement is equivalent to statement 1. If the number of girls is larger than the number of boys by 25% (of the boys,) then girls= (1+25%)boys = 1.25×boys. Don't go into calculating the fraction of girls. If you know the ratio of girls to boys, the situation is similar to that following statement 1. Therefore the correct answer is D. Given a ratio of two parts that add up to the whole, it is possible to determine the fraction of either.
In the Crunchy-Bunchy breakfast cereal, the ratio of oatmeal to corn to wheat used to be 4:3:2. Recently, the Crunchy-Bunchy cereal formula has changed. In the new formula, the ratio of oatmeal to wheat has been divided by three, and the ratio of corn to wheat has been halved. In the new Crunchy-Bunchy formula, what is the ratio of oatmeal to corn? 4:3 1:1 8:9 4:9 2:3 When dividing the ratios, why did we divide only the number on the left, and not both numbers? Remember, a ratio is like a fraction: a ratio of 4:3 is mathematically equivalent to the fraction 4/3. Thus, multiplying or dividing a ratio is like multiplying or dividing a fraction: you need to multiply or divide only the left number (which is equivalent to the numerator), and not both numbers. For example: The ratio 4:3 multiplied by 2 becomes (4×2):3 = 8:3. The ratio 4:3 divided by 2 becomes (4/2):3 = 2:3 Multiplying or dividing both numbers in a ratio is equivalent to expanding or reducing a fraction, not multiplying/dividing it. Just as with fraction, reducing or expanding a ratio doesn't change it's value. This is not the same as multiplying or dividing the ratio, which result in an entirely new ratio. For example, the ratio 2:6 is equal to 1:3 , just as 2/6 is equal to 1/3. To sum this up: The ratio 2:3 multiplied by 5 results in the new ratio 10:3. The ratio 2:3 expanded by 5 results 10:15, which is the same ratio - the change is only cosmetic.
Correct. There are two issues in this question. At first, you must form a new ratio by multiplication/division of another ratio. Then, you should combine the original ratio and the new formed ratio. Break the problem into two steps: First, write down the initial ratio. Go on to divide the oatmeal:wheat ratio by 3, and to halve the corn:wheat ratio. Next, re-combine the ratios by equating the number that represents the common quantity, the wheat. If needed, expand or reduce ratios. Here are the two ratios ...................Oatmeal : Corn : Wheat 1st ratio ..............4..........3........2 Divide the oatmeal:wheat ratio by 3 to get 4/3:2. Halve the corn:wheat ratio to get 3/2:2. Next, combine the two ratios. .......................Oatmeal : Corn : Wheat 1st ratio..............4/3................... 2 ) 2nd ratio........................ 3/2 ......2 ) Since the number representing wheat is already the same number 2, no need to expand/reduce the ratios any further before combining them. The total ratio is 4/3 : 3/2 : 2. Expand by 2·3=6 to get rid of the fractions to get a total ratio of O : C : W of 4·2 : 3·3 : 12 --> 8 : 9 : 12 Thus the ratio of Oatmeal to Corn is 8:9.
Drug Y is the active ingredient in medicine M. Jeff and Zach are being treated with drug Y, which they get exclusively from taking medicine M. If three spoons of medicine M contain 1.75 grams of drug Y, then Zach receives how many grams of drug Y daily? (1) Zach receives half the number of spoons of medicine M as Jeff does. (2) Jeff receives 7 grams of drug Y daily.
Correct. This is a "what is the value of..." DS question. In this type of question, a statement will be sufficient only if it leads to a single value of the variable (or expression) you're asked about. The issue is ratios. The ratio of Medicine M (in teaspoons) to Drug Y (in grams) is given. To find the Real number of grams Zach takes daily, you need the real number of teaspoons of medicine M. Then you can set up a proportion and find the corresponding number of grams of Y in Zach's diet. Stat. (1): without knowing the number of teaspoons Jeff takes, there's no way to find the number of spoons that Zach takes. Stat.(1)->IS->BCE. Stat. (2): might be useful later, but alone, stat. (2) tells you nothing about Zach. Stat.(2)->IS->CE. Stat. (1+2): together, you can find the number of spoons of medicine M imbibed by Zach, and from there you can find the corresponding number of grams of drug Y he takes. Stat.(1+2)->S->C.
A garage has a stock of side mirrors. The ratio of right side mirrors to left side mirrors is 5:3. Iris, a garage worker, attaches pairs of left and right mirrors with an adhesive tape. If 30 right mirrors are left unpaired, how many left and right mirrors are there in the stock? 240 120 80 75 48
Correct. When the problem contains ratios and real numbers, use the ratio box to translate ratios to real numbers. Find out the loophole where you can get the multiplier. It may be the Total row, or, another combination of the rows. The real quantity in the question is the number of unpaired mirrors. Add a row for the unpaired mirrors in the ratio box. Now, how are you going to fill the Ratio column? The ratio units for right mirrors is 5, and for left mirrors it is 3. Therefore, the unpaired=right−left=2. The total is 5+3=8. It follows that the Multiplier is 15, and the Real total is 8×15=120. ................. Ratio ..Multiplier ..Concrete R ...............5.......... L............... 3.......... Unpaired...2.........30/2=15..........30 Total ..........8............*15.......8*15=120
Now that you have encountered problems with two or more ratios, we can discuss the concept of ratio changes. You've already seen that expanding and reducing a ratio doesn't change it at all. a ratio of 1:2 expanded times 2 remains essentially the same ratio - 2:4. To sum up our discussion of ratio change problems: A candy box contains only marshmallows and pralines at a ratio of 2:3. A hungry hippo sneaks in and steals 2 marshmallows, leaving the marshmallows and pralines at a new ratio of 4:9. How many pralines are in the box? When the problem deals with a change in ratio through addition/subtraction i.e., start point ratio, end point ratio, and a real quantity in between: 1) Compare the two ratios - expand / reduce so that the unchanged quantity is represented by the same number in ratio. Example: the original ratio of 2:3 is expanded times 3 to a ratio of 6:9. Note the change in ratio units. ...............................M ..........P Original ratio .....2*3=6... : .3*3=9 change ...............-2..--->-2 marshmallow New ratio............... 4.. : .....9 2) Use the difference in Ratio units and the corresponding change in Real to find the multiplier. Since a drop of 2 in the marshmallow ratio (from 6 to 4) corresponds to a drop of 2 in real (2 marshmallows stolen by the hungry Hippo), the multiplier is simply ×1. 3) Use the multiplier to find the required quantity. Remember to use the ratios in their expanded/reduced form, rather than the original form. The question asks for the number of Pralines, which is simply 9×1 = 9. In Data Sufficiency questions involving ratio changes, a start point ratio, end point ratio, and a real quantity in between are sufficient to answer the question with no calculations needed. In Problem Solving you have to go through all the work.
However, Some GMAT problems introduce the concept of a change in ratio through addition and subtraction. A candy box contains only marshmallows and pralines at a ratio of 2:3. A hungry hippo sneaks in and steals 2 marshmallows, leaving the marshmallows and pralines at a new ratio of 4:9. How many pralines are in the box? Notice the essential ingredients provided by this type of question: 1) An original ratio - 2:3 2) A change (through addition or subtraction) -2 marshmallows 3) A new ratio - 4:9 In order to find the real number of Pralines, we have to find the multiplier. Usually, we'd put the data in a ratio box and find the multiplier by dividing the real numbers presented in the question with the corresponding ratio in the same line. However, the presence of two different ratios makes this difficult. Organize the ratios in a diagram: ...............................M..........P Original ratio .........2... : ... 3 change ....---> -2 marshmallow New ratio............... 4...:......9 The only real number presented by the question is [-2 marshmallows] - the change. The corresponding item must be the change in ratio. However, before we find the change in ratio, the ratios need to be modified so that they are comparable. Right now, these two ratios do not have the same multiplier. Here's the situation right now: ...............................M...........P Original ratio .........2... : ... 3 change ....---> -2 marshmallow New ratio............... 4...:......9 What should you do next? Multiply the original ratio by 2, so that the marshmallows have the same ratio Multiply the original ratio by 3, so that the pralines have the same ratio Multiply the original ratio by 2, so that the marshmallows have the same ratio Incorrect. Recall the lesson on ratio comparison: To combine different ratios you must equate the number representing the member common to both ratios. Since the real number of Marshmallows changes during the question, it is no longer a common member to both ratios. The real number of Pralines is unchanged in both ratios, so this is the common member - multiply the original ratio by 3 to get the same number of ratio units of Pralines. After expanding the original ratio times 3, this is the new situation: ...............................M ..........P Original ratio .....2*3=6... : .3*3=9 change ...............-2..--->-2 marshmallow New ratio............... 4...:......9 Since a drop of 2 in the marshmallow ratio (from 6 to 4) corresponds to a drop of 2 in real (2 marshmallows stolen by the hungry Hippo), the multiplier is simply ×1. The question asks for the number of Pralines, which is simply 9×1 = 9. To further illustrate the problem, here's the the final ratio box of this process. Note that the change in marshmallows gets its own row. .................Combined...Multiplier....Real ..................Ratio Marshmlls.......6 Marshmlls-2...4 change: .....4-6=-2.........*1...............-2 -2 marshmallows Pralines..........9..............*1............9*1=9 Total at present Drawing the box is recommended, but not essential for quickly solving this type of question. It is enough to remember the basic premise - the multiplier is the same for all members of a ratio. Focus on finding the multiplier, and the problem is half solved. What if the question asked about the original number of marshmallows, instead of the new number of marshmallows? Do I use the original ratio of 2:3? Good question. Remember that the multiplier you have found is the same for all ratios in the box. However, the multiplier of ×1 is only relevant to the ratios after the expansion. Thus, If the question asks for the original number of Marshmallows, use the original ratio in its expanded form: not 2:3, but 6:9. Therefore, the original number of marshmallows is 6×1=6.
You've already seen and solved ratio questions using the ratio box. Those questions involved considering a Ratio value with its corresponding Real value. From these two values, you could find the Multiplier, which is the same multiplier for all members of the ratio. The ratio box is an invaluable tool, both as a way to organize the information and as a teaching aid to understand the concept of the multiplier - the connecting factor between ratio and real. The solution - expand or reduce the ratios so that they are comparable. The goal is to get the single member common to both ratios (goats) to the same ratio units (and the same multiplier). Back to our diagram: Camels : Goats : Sheep 1st ratio ( 5 : 2 ) 2nd ratio ( 1 : 3 ) What should you do now? Multiply the first ratio by 3 and the second ratio by 5 Divide the first ratio by 2 Expand the second ratio times two Correct. Multiply both sides of the ratio by two to get 2:6. ...................Camels : Goats : Sheep 1st ratio ( 5 : 2 ) 2nd ratio ( 2 : 6 ) Now that the goats have 2 ratio units in both ratios, the two ratios are comparable. The final ratio of Camels to Sheep is 5:6. To sum up: To combine different ratios you must equate the number representing the member common to both ratios. Write the ratios one on top of the other and expand/reduce the ratios to equate the common member. Example: Turn this ratio pair ....................Camels : Goats : Sheep 1st ratio ( 5 : 2 ) 2nd ratio ( 1 : 3 ) Into this comparable ratio pair: ...................Camels : Goats : Sheep 1st ratio ( 5 : 2 ) 2nd ratio ( 2 : 6 ) So that the Goats have the same ratio units in both ratios.
However, tougher GMAT ratio problems involve several ratios with different multipliers. A petting zoo holds Camels, Goats and Sheep. The ratio of Camels to Goats is 5:2, while the ratio of Goats to sheep is 1:3. What is the ratio of camels to sheep in the petting zoo? This ratio problem requires you to build a third ratio out of two given ratios. To see what's going on, let's organize the information. With two or more different ratios, the following organization works best: ...................Camels : Goats : Sheep 1st ratio ( 5 : 2 ) 2nd ratio ( 1 : 3 ) The question asks for the ratio of Camels to Sheep. Now the average test taker, looking at the diagram above, will jump up and yell "5:3!". And he would, of course, be WRONG. The reason we cannot simply connect the camels and sheep to form a new ratio is that the two ratios are not COMPARABLE. A Ratio is not a Real number - it merely describes a real number. Although the two ratios above describe the same Real number of Camels, Goats and Sheep, they do so in different ways. To see what we mean, suppose that there are 10 Goats in the petting Zoo. Using the Ratio number of Goats and the Real number, we can figure out the Multiplier and the real numbers of the other animals. To illustrate the problem, here are the ratio boxes for both ratios: ..........................CAMELS : GOATS................ ................. Ratio ..Multiplier ..Concrete Cmls ...........5.......... Goats...........2..........10/2=5.........10 Total ...................... .......................GOATS : SHEEP..................... ................. Ratio ..Multiplier ..Concrete Goats...........1..........10/1=10.........10 Sheep...........3 Total ...................... Notice how the same number of goats yields different multipliers for each case (×5 for camels, ×10 for sheep)? Remember the secret of the Ratio Box: The multiplier must be the same for all members of the ratio. This exemplifies why the ratios are not comparable - each "ratio unit" does not represent the same number in Real.
Again, one part fruit extract and 2 parts orange liquor are not real quantities, but rather relative quantities, or ratio. For any real value you choose for one of the ingredients, the other ingredients must comply, so that the ratio is eventually 3 to 2 to 1 or 3:2:1.
To sum up the basics of Ratios: A fraction represents PART/WHOLE, whereas a ratio represents PART/PART . A ratio may be treated like a fraction, i.e., it may be expanded or reduced. Ratios appear in GMAT questions in several formats: The ratio of boys to girls is ¾. The ratio of boys to girls is 3 to 4. The ratio of boys to girls is 3:4. A ratio is not actual amounts - it conveys the relative amounts of each quantity. (e.g. the quantities in a recipe)
Use the ratio box when there is a ratio and a concrete number in the problem. The ratio box translates from ratios to concrete numbers. Set up the ratio box like so: There are always three columns in the following order Ratio, Multiplier, Concrete. Rows are the parts of the ratio that appear in the problem, always followed by the Total row.
You need numbers in the Ratio column and in the Concrete column to figure out the multiplier. The multiplier is applicable for all parts of the ratio box. Once you have it you can get any piece of information the question asks about.
There are three times as many hot dogs as pizzas and five times as many brownies as hot dogs in "Junk Food 'R' Us" restaurant. The restaurant sells only whole products (you cannot buy half a pizza, for example). How many Brownies are there in the restaurant? (1) There are at least 10 pizzas in the restaurant (2) There are 32 hot dogs at most in the restaurant
statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed. Incorrect. This is a "what is the value of..." DS question. The issue is finding the number of brownies. In this type of question, a statement will be sufficient only if it leads to a single value of the variable (or expression) that you're asked about. To combine different ratios you must equate the number representing the member common to both ratios. Write the ratios one on top of the other and expand/reduce the ratios to equate the common member. Note that there are entities that are "unbreakable" - you cannot have half a boy or three quarters of a bus. According to Stat. (1), You can find the ratio of pizzas to hot dogs to brownies, but then what? "at least 10 pizzas" allows for endless trios of real numbers for the three entities. Stat.(1)->IS->BCE. According to Stat. (2), From the question you know two ratios: hot dogs to pizzas and brownies to hot dogs - ..............................H...........P.............B 1st ratio................3...........1 2nd ratio..............1...........................5 Arrange the two ratios so that the common entity - hot dogs - is in the middle: ..............................P...........H.............B 1st ratio................1...........3 2nd ratio............................1............5 To find the combined triple ratio, you have to use the common entity between the two given ratios - the hot dogs. Find the least common multiple for the two different hot dog numbers and expand the ratios accordingly: The least common multiple of 1 and 3 is their product - 3. Thus, you have to expand the second ratio by 3: ..............................P...........H.............B 1st ratio................1...........3 2nd ratio............................3............15 Thus, the ratio of p : h : b is 1 : 3 : 15. It does not help to know that there are 32 hot dogs at most, since it is still possible that the numbers of pizzas, hot dogs and brownies are 1, 3 and 15, or maybe 2, 6, 30, or maybe even 3, 9, and 45. Stat.(2)->IS->CE. According to Stat. (1+2), When constraints are placed on two of the items, namely pizza and hot dogs, it is possible to narrow down the possibilities to one that satisfies both constraints. The only ratio that involves at least 10 pizzas and at most 32 hot dogs is 10:30:150, where there are 150 brownies. Stat. (1+2) -> S.
Remember you can use proportions to solve any problem dealing with a fixed ratio, such as unit conversions (i.e., minutes & seconds, minutes & hours...), the scale of maps, etc.
to sum up: A proportion is an equation with a ratio in fraction format on each side. If one part of the four members of the equation is missing, cross multiply to solve the equation. Make sure the tops of the fractions have the same units, and so do the bottoms of the fractions. Use proportions to solve GMAT problems with fixed ratios, such as maps, unit conversion, etc.
