Free GAMSAT Maths Tips and Formulas

• Angle θ may be given in radians (R) where 1 revolution = 2π^R = 360^°
• Cross-sectional area of a tube = area of a circle = πr² where π can be estimated as 3.14 and r is the radius of the circle; circumference = 2πr.

A.1 Basic Graphs

A.1.1 The Graph of a Linear Equation

Equations of the type y = ax + b are known as linear equations since the graph of y ( = the ordinate) versus x ( = the abscissa) is a straight line. The value of y where the line intersects the y axis is called the intercept b. The constant a is the slope of the line. Given any two points (x₁, y₁) and (x₂, y₂) on the line, we have:

y₁ = ax₁ + b

and

y₂ = ax₂ + b.

Subtracting the upper equation from the lower one and dividing through by x₂ - x₁ gives the value of the slope, a = (y₂ - y₁)/(x₂ - x₁).

Note: a positive slope rises as it extends to the right (as in the graph below), a negative descends, and if the line is horizontal, then the slope is zero.

A.1.2 Reciprocal Curve

For any real number x, there exists a unique real number called the multiplicative inverse or reciprocal of x denoted 1/x or x^-1 such that x (1/x) = 1. The graph of the reciprocal 1/x for any x is:

A.1.3 Miscellaneous Graphs

There are classical curves which are represented or approximated iin the Gold Standard GAMSAT textbook as follows (if you do not have the book, we suggest doing a Google image search so you can identify these shapes of these graphs/curves): Sigmoidal curve (CHM 6.9.1, BIO 7.5.1), sinusoidal curve (PHY 7.1.1, 7.1.2), and hyperbolic curves (CHM 9.7 Fig III.A.9.3, BIO 1.1.2).

If you were to plot a set of experimental data, often one can draw a line (A.1.1) or curve (A.1.2/3, A.2.2) which can "best fit" the data. The preceding defines a regression line or curve. The main purpose of the regression graph is to predict what would likely occur outside of the experimental data.

A.2 Exponents and Logarithms

A.2.1 Rules of Exponents

• a⁰ = 1
• a¹ = a
• aⁿ a^m = a^n+m
• aⁿ/a^m = a^n-m
• (aⁿ)^m = a^nm
• 

A.2.2 Exponential and Logarithmic Curves

The exponential and logarithmic functions are inverse functions. That is, their graphs can be reflected about the y = x line.

Figure A.1: Exponential and Logarithmic Graphs. A > 0, A ≠ 1.

A.2.3 Log Rules and Logarithmic Scales

The rules of logarithms were discussed in detail in GAMSAT Maths (GM 3.7, 3.8) and in context of Acids and Bases in General Chemistry (CHM 6.5.1). These rules also apply to the "natural logarithm" which is the logarithm to the base e, where "e" is an irrational constant approximately equal to 2.7182818. The natural logarithm is usually written as ln x or log_e x . In general, the power of logarithms is to reduce wide-ranging numbers to quantities with a far smaller range.

For example, the graphs commonly seen in the Gold Standard GAMSAT textbooks, including the preceding one, are drawn to a unit or arithmetic scale. In other words, each unit on the x and y axes represents exactly one unit. This scale can be adjusted to accommodate rapidly changing curves. For example, in a unit scale the numbers 1 (= 10⁰), 10 (= 10¹), 100 (= 10²), and 1000 (= 10³), are all far apart with varying intervals. Using a logarithmic scale, the sparse values suddenly become separated by one unit: Log 10⁰ = 0, log 10¹ = 1, log 10² = 2, log 10³ = 3, and so on.

In practice, logarithmic scales are often used to convert a rapidly changing curve (e.g. an exponential curve) to a straight line. It is called a semi-log scale when either the x-axis or the y-axis is logarithmic. It is called a log-log scale when both x-axis and y-axis are logarithmic.

Many GAMSAT problems every year rely on a basic understanding of logarithms for pH problems, rate law (CHM 9.10) or a 'random' Nernst equation question (BIO Chapter 0, GS-1). Here are the rules you must know:

1. log_aa = 1
2. log_aM^k = k log_aM
3. log_a(MN) = log_aM + log_aN
4. log_a(M/N) = log_aM - log_aN
5. 10^log₁₀^M = M

For example, let us calculate the pH of 0.001 M HCl. Since HCl is a strong acid, it will completely dissociate into H⁺ and Cl^-, thus :

• [H⁺] = 0.001
• -log[H⁺] = -log (0.001)
• pH = -log(10^-3)
• pH = 3 log 10 (rule #2)
• pH = 3 (rule #1, a = 10)

A.3 Simplifying Algebraic Expressions

Algebraic expressions can be factored or simplified using standard formulae:

• a(b + c) = ab + ac
• (a + b)(a - b) = a² - b²
• (a + b)(a + b) = (a + b)² = a² + 2ab + b²
• (a - b)(a - b) = (a - b)² = a² - 2ab + b²
• (a + b)(c + d) = ac + ad + bc + bd

A.4 Properties of Negative and Positive Integers

Here are some challenging math problems, but if you can apply the basic rules then real GAMSAT math (i.e. related to physics and general chemistry problems) will not give you difficulties. The questions are followed by answers and worked solutions/explanations.

1) Simplify the expression: (x²)(y²)(x³)(y)(x⁰).

1. x⁶y³

2. x⁵y³

3. x⁵y²

4. xy³

5. 0

2) Simplify the expression: (2x)^-2(((2x²)³)²).

1. 16x⁵

2. 16x⁶

3. 16x⁸

4. 16x¹⁰

5. 16x¹²

3) Expand the expression: (x - y + 3)².

1. x² - y² + 9

2. x² + y² + 9

3. x² + y² + 2xy + 6x + 6y + 9

4. x² + y² - 2xy + 6x - 6y + 9

5. x² - y² - 2xy + 6x - 6y + 9

4) Simplify the expression: (x^{2a + b})(x^{a - 2b}) / (x^{2a - b}).

1. x^a

2. x^ab

3. x^{a + 2b}

4. x^{5a - 2b}

5. x^{5a + 2b}

5) Let x = 4 and y = 8. Evaluate the expression: ((y^-2/3)^1/2) / (x^-1/2).

1. 8

2. 4

3. 1

4. 1/2

5. 1/4

6) Evaluate the expression: log₆(24) + log₆(9).

1. 3

2. 2

3. 1

4. 1/2

5. 1/3

7) Solve for x: log₁₀(70) = x + log₁₀(7).

1. 0

2. 1

3. 2

4. 3

5. 4

8) Simplify the expression: x(log_b(y)) + y(log_b(y)).

1. log_b(y^x-y)

2. log_b(y^x+y)

3. log_b(y^xy)

4. log_b(xy^xy)

5. log_b(x^y)

9) Evaluate the expression: ln (e³)log₃(27) + ln (1)ln (e).

1. e

2. 3

3. 6

4. 3e

5. 9

10) Solve for b: log_b(9) - 2(log_b(12)) = 2.

1. ¼

2. ½

3. 1

4. 2

5. 4

11) Which equation matches the graph below?

1. y = 2^x - 1

2. y = -(2^x)+ 1

3. y = 2^x

4. y = -(2^x)

5. y = -(x²)

12) Which equation matches the graph below?

1. y = ln (x)

2. y = -ln (x - 1)

3. y = -ln (x + 1)

4. y = ln (x - 1)

5. y = ln (x + 1)

13) As x increases, the slope of f(x) = 2^x-1:

1. decreases.
2. increases.
3. remains constant.
4. sometimes decreases, sometimes increases.
5. Not Enough Information

14) pH is measured on a logarithmic scale given by the equation pH = - log₁₀(H), where 0 < H <1. As H decreases, the slope of the graph of pH vs H:

1. Decreases
2. Increases
3. Remains Constant
4. Sometimes Decreases, Sometimes Increases
5. Not Enough Information

1. B
2. D
3. D
4. A
5. C
6. A
7. B
8. B
9. E
10. A
11. D
12. E
13. B
14. B

1) First rearrange the expression and group like terms:

(x²)(y²)(x³)(y)(x⁰) = (x² x³ x⁰)(y²y)

When multiplying powers with the same base, you can combine them by adding the exponents. For example:

y²y = (y)(y)(y) = y²⁺¹ = y³

So the expression becomes:

(x² x³ x⁰)(y²y)

= x²⁺³⁺⁰y²⁺¹

= x⁵y³

2) We can break this problem two chunks: (2x)^-2 and (((2x²)³)²). First consider (2x)^-2. Because the exponent is negative, this is equal to the inverse of the positive power.

(2x)^-2 = 1/(2x)²

Next, distribute the exponent throughout the parenthetical expression:

1/(2x)² = 1/(2²x²)

Now lets move to our second chunk, (((2x²)³)²). When you raise a power to another power, multiply the exponents together. And don't forget to distribute through the parenthetical expression.

(((2x²)³)²) = ((2³x⁽²⁾⁽³⁾)²) = (2⁽³⁾⁽²⁾x^(2)(3)(2)) = 2⁶x¹²

Finally, let's multiply the two chunks together to get back to the original expression:

(2x)^-2(((2x²)³)²)

= [1/(2²x²)](2⁶x¹²)

= (2⁶x¹²)/(2²x²)

When dividing powers, subtract the exponents:

= (2^6-2x^12-2)

= 2⁴x¹⁰

= 16x¹⁰.

3) When raising a polynomial to a power, treat everything inside the parentheses as if it was a single value. So (x - y + 3)² is really (x - y + 3)(x - y + 3), NOT (x² - y² + 3²). Now let's multiply and expand:

(x - y + 3)(x - y + 3)

= x² - xy + 3x - xy + y² -3y + 3x - 3y + 9

= x² + y² - 2xy + 6x - 6y + 9

4) First simplify the numerator:

(x^{2a + b})(x^{a - 2b}) / (x^{2a - b})

= (x^2a+b+a-2b) / (x^{2a - b})

= (x^{3a - b}) / (x^{2a - b})

Then combine the numerator and denominator using the properties of exponent division.

= (x^3a-b-2a+b)

= x^a

5) First combine the exponents where possible, and rearrange so they are all positive:

((y^-2/3)^1/2) / (x^-1/2)

= (y^-1/3) / (x^-1/2)

= (x^1/2) / (y^1/3)

Now plug in x = 4 and y = 8. Notice that 4 = 2² and 8 = 2³.

= (4^1/2) / (8^1/3)

= (2^(2)1/2) / (2^(3)1/3)

= 2¹/2¹

= 1.

6) When adding logarithms of the same base, combine them by multiplying the numbers in parentheses. In this case:

log₆(24) + log₆(9)

= log₆(24*9)

= log₆(216)

= log₆(6³)

Now remember, a logarithm is an exponent. The question it poses is, "the base raised to what power is equal to the number in the parentheses?" So 6 raised to what power is equal to 6³? The answer is, of course, 3.

7) First isolate the x terms on one side and all other terms on the other side of the equation.

log₁₀(70) = x + log₁₀(7)

log₁₀(70) - log₁₀(7) = x

Now combine the logarithms. When subtracting logarithms of the same base, combine them by dividing the numbers in parentheses.

log₁₀(70/7) = x

log₁₀(10) = x

1 = x.

8) A coefficient multiplied by a logarithm can by brought inside the parentheses as an exponent.

x(log_b(y)) + y(log_b(y))

= log_b(y^x) + log_b(y^y)

= log_b(y^xy^y)

= log_b(y^x+y).

9) The natural log has base e. Note that a logarithm of 1, no matter what the base, is equal to 0. And a log of its own base is equal to 1. So:

ln (e³)log₃(27) + ln (1)ln (e)

= ln (e³)log₃(27) + 0*1

= 3 log₃(27)

= 3(3)

= 9.

10) First simplify the logarithms on the left side.

log_b(9) - 2(log_b(12)) = 2

log_b(9) - log_b(12²) = 2

log_b(9/144) = 2

log_b(1/16) = 2

Now convert the equation into exponent form using the definition of logarithms.

b² = 1/16

b = √(1/16)

b = ¼.

11) There are some useful pieces of information to notice that will help you answer this type of problem.

- What is the x and/or y intercept, if there is one?

- Where is the vertical asymptote? [Note: an 'asymptote' refers to a line that keeps approaching a given curve but does not meet the curve at any finite distance.] And does the curve approach positive or negative infinity?

In this case there is no x intercept, but the y intercept is at the point (0, -1). Plugging x = 0 into the given equations we can rule out all options except y = -(2^x), so that is the solution.

12) There are some useful pieces of information to notice that will help you answer this type of problem.

- What is the x and/or y intercept, if there is one?

- Where is the vertical asymptote? And does the curve approach positive or negative infinity?

The x and y intercept of this graph are the same, at (0, 0). Plugging in y = 0 to the given equations we can eliminate all but y = ln (x+1) and y = -ln (x+1). Next find the vertical asymptote. It appears to be located at x = -1, and the curve approaches negative infinity. When x is small, -ln (x+1) is positive, so it cannot be the solution. Thus the graph represents y = ln (x+1).

13) The -1 in the exponent of f(x) = 2^x-1 does not change the behavior of the slope, it simply shifts the graph along the x-axis. The slope of f(x) = 2^x increases exponentially as x increases.

14) You can think of pH and H as corresponding to y and x respectively. So the graph you are considering is y = -log₁₀(x), the logarithm graph reflected about the x-axis. So the slopes along the curve are the opposite of the positive logarithm graph. Therefore when we decrease the value of x (moving right to left along the axis) the slope increases. If you are unsure of your solution, plug in test points to check.

Discuss any of our GAMSAT maths questions or worked solutions here: GAMSAT Math Forum

If you come from a non-science background, you can find our preparation advice here: Non-Science Background: GAMSAT Preparation

Get 1 full hour access to free GAMSAT Physics and GAMSAT Chemistry and Biology videos online. Register here: Free GAMSAT-prep User Account

Gold Standard's GAMSAT Guide

To help you prepare for the exam, Gold Standard GAMSAT has assembled comprehensive information on GAMSAT scores, topics covered, preparation advice, free GAMSAT sample questions and other study resources.

• What is the GAMSAT? - GAMSAT-prep.com's detailed guide about the GAMSAT
• GAMSAT Scores - A complete resource that explains how the GAMSAT is scored and other questions relating to GAMSAT scoring
• Free GAMSAT Resources - A comprehensive guide on preparing for each section of the GAMSAT with links to relevant study resources
• Australian Medical Schools Admission Requirements - A comprehensive list of GAMSAT scores and GPA scores for graduate entry medicine (GEM) programmes in Australia and Australasia
• Free GAMSAT Study Schedule - Even if you don't own any of the Gold Standard GAMSAT materials, this study schedule remains helpful as it details the subjects covered for review and practice on a daily basis.
• GAMSAT Study Tips on Efficient Note-taking - One of our popular resources! Read tips on how to consolidate your review by creating brief notes a.k.a. Gold Notes that will be vital in your daily review in the lead up to the real exam.
• Free Monthly GAMSAT Webinars - Take advantage of our free monthly online seminars (webinars) using a problem-based learning approach on the different topics in the GAMSAT and get a chance to win a GAMSAT giveaway.
• Non-science Background GAMSAT Preparation - Dr Ferdinand (author of The Gold Standard GAMSAT books) outlines the 70/30 formula to get a high score. Also includes a link to advice on related topics.

Free GAMSAT practice:

• Free GAMSAT Practice Test - This abbreviated, timed practice test includes all three sections with instant scaled scores and helpful worked solutions.
• GAMSAT Practice Question of the Day - Develop your GAMSAT reasoning skills with daily practice, one subject at a time
• GAMSAT Physics Sample Questions - 15 free GAMSAT Physics sample questions with answers and worked solutions to hone your basic problem-solving skills and understanding of Section 3 material