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GMAT Tips: Prop...

Exams Know-how

GMAT Tips: Proportions and Ratios

Q: What are proportions and how are they represented?

Proportions are equations that express the equality of two ratios. They are represented using the symbols '::' or '='. For example, if the ratio of x to y equals the ratio of a to b, it can be written as x:y=a:b or x:y::a:b.

Q: How can understanding ratios help in solving GMAT questions?

Understanding ratios is crucial for solving GMAT questions because it allows you to compare different quantities effectively. Ratios simplify the relationship between numbers, making it easier to manipulate and solve problems related to mixture, speed, work, and other quantitative areas tested in the GMAT.

Q: What is the importance of maintaining the order of terms in ratios?

Maintaining the order of terms in ratios is important because the meaning of the ratio depends on the order. For example, a ratio of 2:3 is different from 3:2. The first term (antecedent) and the second term (consequent) must be placed correctly to represent the relationship accurately.

Q: How can cross multiplication be used to solve proportion problems?

Cross multiplication is a method to solve proportion problems by eliminating the fractions. If you have a proportion a/b = x/y, cross multiplying gives a⋅y=b⋅x. This equation can then be solved for the unknown variable, making it easier to find the solution.

Q: What strategies can solve mixture problems involving ratios and proportions?

To solve mixture problems, identify the quantities of each component and set up equations based on the given ratios. The total amount needed for the desired ratio is determined and solved for the unknown. Use methods such as setting up proportions, cross-multiplying, and adjusting the components to achieve the required concentration.

Yocket Editorial Team

Jul 10, 2024

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You must take the GMAT to gain admission to a top-ranked business school. A global exam, GMAT assesses your ability to perform well in quant and reasoning. One integral part of the exam is proportion and ratios. You may lose your marks if you skip this portion!

Do you need clarification on how to solve Proportion and Ratio questions? Do you often get incorrect answers because the basics of the topic are unclear? If yes to any of the above questions, read on to understand how to solve GMAT Proportion and Ratios questions.

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Table of Contents

Concept 1: Proportion
Concept 2: Ratios
Properties for Proportion and Ratio Questions
GMAT Proportion and Ratio Examples
Tips on How to Solve GMAT Proportion and Ratios Questions
From the Desk of Yocket

Concept 1: Proportion

The proportion is represented by the symbols '::' or '='. If the x:y ratio equals the a:b ratio, then x, y, a, and b are proportional. It is represented by the symbols x:y=a:b or x:y: a:b. When four words are proportional, the product of two intermediate values (the second and third) must equal the product of two extremes.

Concept 2: Ratios

The ratio is the simplest form or comparison of two related numbers. A ratio is a number that expresses one quantity as a fraction of another. For example, the ratio between the integers 5 and 6 is 5:6. It also shows the number of times one quantity equals another. The "terms" refer to the numbers that comprise the ratios. The upper part of the ratio (numerator) is referred to as the antecedent, while the lower part of the proportion (denominator) is known as the consequent or descendent. For example, if the ratio is 4:6, 4 is the antecedent and 6 is the descendent.

Properties for Proportion and Ratio Questions

Students must consider the following important properties when solving ratio and proportion questions.

Property Name	Property Function
Componendo and dividendo	If x:y=a:b then x+y:x–y=a+b:a-b
Invertendo	If x:y=a:b, then y:x=b:a
Alternendo	If x:y=a:b then x:a=y:b
Componendo	If x:y=a:b then x+y:x=a+b:a
Dividendo	If x:y=a:b then x-y:a=a-b:a
Subtrahendo	If x:y=a:b then x-a:y-b
Addendo	If x:y=a:b, then x+a:y+b

GMAT Proportion and Ratio Examples

Let us understand the concept of proportion and ratios with the help of some of the examples.

Problem 1:

A sum of Rs. 6935 is divided among Alice, Bob, and Charlie such that if Rs. 35, Rs. 45, and Rs. 30 are deducted from their respective shares, they have money in the ratio 5:7:9. What is Alice’s share?

A. 1475

B. 1565

C. 1660

D. 1725

Solution:

If we deduct 35, 45, and 30 rupees from Alice’s, Bob’s, and Charlie’s shares, their resulting amounts are in the ratio 5:7:9.

Let their reduced shares be 5x, 7x, and 9x. Therefore:

Alice’s share = 5x + 35,

Bob’s share = 7x + 45,

Charlie’s share = 9x + 30.

The total amount is:

(5x + 35) + (7x + 45) + (9x + 30) = 6935.

21x = 6935 - 110 = 6825.

Thus, x = 6820 / 21 = 325.

Alice’s share is 5x + 35 = 5*325 + 35 = 1625 + 35 = 1660.

Answer: C

Takeaway:

When given a ratio problem with specific amounts to be deducted, set up equations with the given ratios and solve for the unknown variable. This approach simplifies the problem and ensures accuracy.

Problem 2:

If 12 workers can paint 12 walls using 12 brushes in 12 days, how many days will it take 9 workers to paint 9 walls using 9 brushes?

A. 9 days

B. 10 days

C. 12 days

D. 13.5 days

Solution:

12 workers in 12 days means 144 worker-days are required to paint 12 walls.

1 wall requires 12 worker days. Thus, 9 walls will need 108 worker days.

With 9 workers, the time required is 108 worker-days / 9 workers = 12 days.

Answer: C

Takeaway:

Convert complex work problems into simpler units like worker days to understand the total effort required, and then use this to find the solution efficiently.

Problem 3:

Three quantities P, Q, and R are such that PQ = kR, where k is a constant. When P is fixed, Q varies directly as R; when Q is fixed, P varies directly as R, and when R is fixed, P varies inversely as Q.

Initially, P was at 4, and P:Q: R was 2:5:6. Find the value of P when Q equals 10 at constant R.

A. 5

B. 6.67

C. 10

D. 8

Solution:

Initial values are 4, 10, and 12.

Thus we have

PQ=kR

4 * 10 = k * 12.

Hence, k = 40 / 12 = 10/3.

Thus, the equation is PQ = (10/3)R.

For the problem, keep R constant at 12. Then, P * 10 = (10/3) * 12.

i.e., P = (120 / 10) = 8.

Answer: D

Takeaway:

Understanding the relationships between variables (direct or inverse variation) allows you to manipulate the given equations to find unknown values.

Problem 4:

If p/q = 5/6. Find the value of the expression, (7p – 4q)/(8p + 3q).

A. 2/11

B. 5/33

C. 6/43

D. 7/48

Solution:

Assume the values as p = 5 and q = 6.

Then we have:

(75 - 46) / (85 + 36) = (35 - 24) / (40 + 18) = 11 / 58 = 11/58.

Answer: B

Takeaway:

When dealing with expressions involving ratios, substituting simple values that maintain the ratio can make the calculation straightforward and manageable.

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Problem 5:

The ratio of sand to cement in a mixture is 3:1. Find the quantity of sand to be added to the mixture to make this ratio 5:1 if the current cement quantity remains the same at 4 kg.

A. 4 kg

B. 6 kg

C. 8 kg

D. 10 kg

Solution:

Currently, the mixture has 3 parts sand and 1 part cement. We have 12 kg of sand for 4 kg of cement.

To achieve a 5:1 ratio, the amount of sand required is 5 times the amount of cement:

Thus, required sand = 5 * 4 = 20 kg.

Extra sand to be added = 20 kg - 12 kg = 8 kg.

Answer: C

Takeaway:

Keep one component constant and adjust the other to achieve the desired ratio. This is a common strategy for mixture problems.

Problem 6:

Three containers X, Y, and Z contain a mixture of oil and water in ratios 2:3, 3:4, and 4:5 respectively. If the capacities of the containers are in the ratio 6:5:4, find the ratio of oil to water when the mixtures from all three containers are combined.

Solution:

Assume the capacities of X, Y, and Z as 600, 500, and 400 liters respectively.

Then the amounts of oil are:

X: (2/5) * 600 = 240 liters,

Y: (3/7) * 500 = 214.29 liters,

Z: (4/9) * 400 = 177.78 liters.

Total oil = 240 + 214.29 + 177.78 ≈ 632.07 liters.

Total capacity = 600 + 500 + 400 = 1500 liters.

Total water = 1500 - 632.07 ≈ 867.93 liters.

The ratio of oil to water = 632.07: 867.93 = 0.7275: 1 ≈ 53: 73.

Answer: 53:73

Takeaway:

When combining ratios from different sources, calculate the total amount of each component separately before finding the final ratio.

Problem 7:

If 80 ml of a solution contains 16% acid, how much water must be added to create a solution with 10% acid?

Solution:

Let x ml of acid be present in the solution.

Then,

16/100 = 𝑥/80 → 𝑥 = 12.8 ml.

Therefore, 12.8 ml of acid is present in 80 ml of solution.

To dilute this to a 10% solution, let the final volume of the solution be y ml.

Then, 10/100 = 12.8/𝑦 → 𝑦= ml.128

So, to get a 10% acid solution, we need to add 128-80 = 48 ml of water.

Takeaway:

When diluting a solution, calculate the final volume for the desired concentration and subtract the original volume to find the amount of water to add.

Problem 8:

A 25-liter solution contains 30% alcohol. How much alcohol needs to be added to make it a 50% alcohol solution?

Solution:

The initial solution has 30% alcohol, which means there are 7.5 liters of alcohol (0.30 * 25 = 7.5).

Let x liters of alcohol be added.

The new total volume of the solution will be 25+𝑥 and the new amount of alcohol will be 7.5+x liters.

To make it 50% alcohol, the equation is:

(7.5+𝑥)/(25+𝑥)=50/100

Solving,

7.5+𝑥=0.5(25+𝑥)

7.5+𝑥=12.5+0.5𝑥

0.5𝑥=5

x=10

So, 10 liters of alcohol need to be added.

Takeaway:

To increase the concentration of a solution, set up a proportion equation based on the desired concentration and solve for the amount of substance to be added.

Tips on how to solve GMAT Proportions and Ratios Questions

Below are some tips that can be used while solving GMAT Proportions and Ratios questions.

Same Kind Comparison: Ensure that the ratio between two quantities exists with the same type of measurement (e.g., length to length, weight to weight).
Similar Units: When comparing two ratios, their units must be the same.
Order of Terms: Maintain the significant order of terms while dealing with ratios.
Fraction Comparison: Ratios can only be compared if represented as fractions.
Cross Multiplication: If a/b=x/y, then ay=bx
Proportion Interchange: If a/b=x/y, then a/x=b/y
Reciprocal Ratios: If a/b=x/y, then b/a=y/x
The sum of Ratios: If a/b=x/y, then (a+b)/b=(x+y)/y
Difference of Ratios: If a/b=x/y, then (a-b)/b=(x-y)/y
Sum and Difference: If a/b=x/y, then (a+b)/(a-b)=(x+y)/(x-y)
Equal Ratios: If x/(y+z)=y/(z+a)=z/(x+y) and x+y+z is not equal to 0 then x=y=z

From the Desk of Yocket

GMAT Proportions and Ratios seem complex, but you can clear it with the right strategy and preparation. You can grasp the topic by starting with simple questions on Proportions and Ratios and then upgrading your preparation. The above tips will help you understand ways to solve these questions easily.

If you need more clarity on GMAT, speak with a Yocket expert. With 15+ years of experience, the study-abroad experts at Yocket are here to help you. Whether it is about the assistance in GMAT, or about which universities will be the best fit for you - Yocket has the right strategy to find out all. Explore Yocket Premium to know more.