Intro, Strategy & Math (Altius MCAT Prep)

Estimating Fractions:

For fractions where the numerator is larger, create a compound fraction. For example, 13/5 becomes 2 3/5 or exactly 2.60. For fractions where the denominator is larger, try the "high/low" method. Change the denominator to one digit higher and to one digit lower. In most cases this will yield at least one familiar fraction. For example, 3/7 is changed to 3/6 and 3/8. It is thus a little less than 0.5, say 0.45. Similarly, 7/13 is changed to 7/12 and 7/14. Thus it is a little more than 0.5.

What is the decimal equivalent of 16/3?

Since the denominator is smaller on this fraction, the best method to use is to convert it to a compound fraction. 16/3 becomes compound fraction 5 1/3 = 5.333

What is the cube root of 25?

The cube root of 8 is 2 and the cube root of 27 is 3. Remember cube root of 8 is the same as (2 x 2 x 2) and the cube root of 27 is the same as (3 x 3 x 3). So the cube root of 25 has to be in between 2 and 3 and since 25 is closer to 27 it will probably be 2.9~

Change 2/7 to a decimal.

Use the "high/low" method here because the denominator is larger. 2/6 can be reduced to 1/3, which equals 0.33 2/8 can be reduced to 1/4, which equals 0.25 So, 2/7 is approximately 0.29

Q5. All of the angles in any triangle must add up to: ___________

180 degrees

What is the compound fraction of 13/5 ?

2 3/5 or exactly 2.60

Q3. How do you simplify if you end up with a number like: 0.05 x 10^-2 or 220 x 10^12?

A number such as 0.05 x 10-2 can be simplified by moving the decimal to the right or to the left to achieve the correct arrangement with only one digit prior to the decimal. The mnemonic LARS can be used to remember that moving to the Left requires adding to the exponent and moving to the right requires Subtracting from the exponent (Left = A; R = Subtract = LARS). Please NOTE that when it is said one should "add" or "subtract" it must be applied in a literal algebraic sense, NOT in terms of the magnitude of the exponent. For example if moving the decimal two places to the left, one should add 2 units to the exponent. If the exponent is 10^-4 it does NOT become 10^-6, it becomes 10^-2 (because adding 2 to -4 gives -2). Similarly, if one moves the decimal two places to the right one should subtract 2 units from the exponent. If the exponent is 10^-4 it becomes 10^-6 (because subtracting 2 from -4 gives -6).

What is the Sin of 0 , 30 , 45, 60 , and 90 degrees?

0 degrees = 0 30 degrees = 1/2 45 degrees = square root of 2 /2 = (0.7) 60 degrees = square root of 3/2 = (0.9) 90 degrees = 1

What is the Cos of 0 , 30 , 45 , 60 , 90 degrees?

0 degrees = 1 30 degrees = square root of 3/2 = (0.9) 45 degrees = square root of 1 /2 = (0.7) 60 degrees = 1/2 90 degrees = 0

What is the square root of 2/ 2 equal to?

0.7

What is the square root of 3/ 2 equal to?

0.9

What is the square root of 2 equal to?

1.4

What is the square root of 3 equal to?

1.7

What are the decimal equivalents of 1/2 , 1/3 , 1/4 , 1/5 , and 1/8 ?

1/2 = 0.5 1/3= 0.333 1/4= 0.25 1/5= 0.2 1/8=0.125

What is the square root of 120?

10 squared equals 100 and 11 squared equals 121 , so the square root of 120 should equal approximately 10.9~ .

Q4. How do you write a small whole number, such as 5, in scientific notation?

Any number can be written as the number itself times 100. This is possible because 100 is equal to one. In fact, you should know that any number raised to the zero power is one. It is also important to recall that any number raised to the first power is the number itself.

Other Important Trigonometry Formulas:

Area of a circle: A = πr^2 Circumference of a circle: C = πd or 2πr Area of a triangle: A = 1/2bh Volume of a sphere: V = 4/3πr^3 Surface Area of a sphere: SA = 4πr^2 Pythagorean Theorem: A^2 + B^2 = C^2

Estimating Square Roots:

It is first helpful to know the square of all numbers from 1 through 15. Commit these squares to memory if you haven't already. Once these are committed to memory, start squaring numbers. Find one you know that is just more than the number you are taking the square root of, and one that is just less. The answer will be in between those two. For example, the square root of 72 is estimated by saying that 8^2 is 64 and 9^2 is 81. Seventy-two is approximately in the middle of 64 and 81, so the square root of 72 is about 8.5.

What is the decimal equivalent for 3/13?

Use the "high/low" method here because the denominator is larger. 3/12 = 0.25 ~ 3/14= 0.21 ~ So, 3/13 is approximately 0.23

Change 24/75 to a decimal.

Use the "high/low" method here because the denominator is larger. 24/74 = 0.32~ 24/76= 0.31~ 24/75 has to equal approximately 0.3~ . Remember you don't need to be very specific.

Q1. Practice your estimation skills by completing the following calculations in your head: a) What is the square root of 75? b) Change 24/45 to a decimal. c) What is the square root of 120? d) What is the decimal equivalent for 3/13? e) Change 2/7 to a decimal. f) What is the decimal equivalent of 16/3? g) What is the cube root of 25? Check your answers using a calculator.

a) To estimate the square root of 75, imagine a number squared that is close to, but just below 75. The closes would be 8^2, which equals 64. Next, square the next integer: 9^2, which equals 81. Because 75 is 9 more than 64 and 6 less than 81, the square root of 75 must be between 8 and 9, but closer to 9, about 8.6 [actual = 8.66]. b) To change 24/45 to a decimal, change 45 to 48: 24/48 = 0.5. Because 48 is larger than 45 the real answer is slightly more than 0.5, say 0.55 [actual = 0.533]. c) To estimate the square root of 120 use the same lower/higher method used previously. 10^2 is 100, and 11^2 is 121. Because 121 is so close to 120, we know the square root of 120 is essentially 11, or probably 10.97 [actual = 10.95]. d) To change 3/13 to a decimal, use the high/low method: 4/13 and 2/13 or 3/12 and 3/14. It should jump out to you that 3/12 reduces to 1⁄4 and thus 3/13 is slightly less than 1⁄4, say 0.23 [actual = 0.23]. e) To change 2/7 to a decimal use the high-low method: 1/7 or 3/7 and 2/6 or 2/8. Because 2/6 is exactly 1/3 and 2/8 is exactly 1⁄4, we know that 2/7 is in the middle of 0.33 and 0.25, say 0.29 [actual = 0.286]. f) To change 16/3 to a decimal, first change it to a complex fraction by dividing 3 into 16 and putting the remainder over 3: this gives 5 1/3, or 5.33 [actual = 5.33]. g) To estimate the cube root of 25, cube numbers until you find one just less than 25. 2^3 = 8; 3^3 = 27, which is very close to 25, so the cube root of 25 is just less than 3, say 2.9 [actual = 2.92].

Q2. What do you do to the exponents when you perform the following functions on numbers in scientific notation? a) add/subtract, b) multiply, c) divide, d) raise something in scientific notation to a power, e) take the square (or other) root of something in scientific notation.

a) add = you must change the number to make the exponents the same. You can only add or subtract numbers in scientific notation if they have the same exponent. Once the exponent is the same, SUBTRACT the mantissa as you would any other number and then give the answer the same exponent. For example, 2.1 x 10^3 subtracted from 3.5 x 10^4 first requires changing 3.5 x 10^4 to 35 x 10^3. The problem becomes 35 x 10^3 minus 2.1 x 10^3. We can subtract 2.1 from 35 to give 32.9 x 10^3 (or 3.29 x 10^4). b) add the exponents, c) subtract exponent in the denominator from the exponent in the numerator, d) multiply the exponents AND raise the mantissa to the power. For example: (2 x 10^4)2 = 4 x 10^8 ; it does NOT equal 2 x 10^8. e) raising a number in scientific notation to the 1⁄2 power is the same as taking the square root of that number, raising it to the 1/3 power is the same as taking the cube root of it, and so forth. One would then take the multiply the exponents. For example (2 x 10^4)^1/2 = 1.4 x 10^2.

Independent Variable

is the variable that an experimenter manipulates or controls in an experiment. It may also be a variable through which the experimenter groups participants.

Dependent Variable

is the variable that the experimenter believes will be effected by the independent variable, and is measured in some way. The dependent variable varies as a function of the independent variable.

Trigonometric Relationships:

sin θ,=O/H; cos θ,=A/H; tan θ,=O/A cosecant, secant and cotangent are the inverses of sin, cos and tan, respectively. tan θ, = sin θ/cos θ sin^2x + cos^2x=1

Converting from degrees to radians:

π radians = 180°; 2π radians = 360°. There are approximately 6 radians in one circle. Thus, if something is rotating at 12 rad/s, you know that it is making two revolutions per second.

Commit to memory the following approximate decimal equivalents related to the sin and cosine of common angles:

√2 = 1.4; √3 = 1.7; √2/2 = 0.7; √3/2 = 0.9