1/3 in decimal is 0.333...

Using logical reasoning:

What is 1/9 in decimal?

What is 7/9 in decimal?

List 2 ways of comparing fractions

SQUARING a positive proper fraction/percent INCREASES/DECREASES the value?

DECREASES

e.g. 1/4 x 1/4 = 1/16

What are the 2 "percent change" equations?

ORIGINAL + CHANGE = NEW

CHANGE/ORIGINAL = PERCENT CHANGE

If a quantity is increased by x percent, then what, in algebraic terms, is the new quantity as a percent of the original?

The new qty. is (100 + x)% of the original... i.e. a 15% increase produces a quantity that's 115% of the original...

I.E. ORIGINAL*(1 + PCT INCREASE/100 ) = NEW

If a quantity is decreased by x percent, then what, in algebraic terms, is the new quantity as a percent of the original?

The new qty. is (100 - x)% of the original... i.e. a 15% decrease produces a quantity that's 85% of the original...

I.E. ORIGINAL*(1 - PCT INCREASE/100 ) = NEW

A 20% decrease in the price of a $30 shirt results in a new price of $24.

What percentage of the old price is the NEW price?

24/30 = 4/5 = x/100

... 5x = 400 therefore x = 80

therefore the new price is 80% of the original price

i.e. new price = 0.8(old price)

What is x% of y, written algebraically?

(x/100)y

A part is come percent of a whole.

Write this as an equation

PART/WHOLE = PERCENT/100

Write an easy way to find 10% of any number?

What's a good use of this technique?

1) Just move the d.p. to left one place

e.g. 10% of 24.99 = 2.499

2) You can use 10% as a benchmark value for estimating percents.

For Data Sufficiency problems involving percent change, all you need to compute a percent change is ____ ?

The RATIO of ANY TWO of the following: Original, Change and New

In the percent change formula, ORIGINAL +- CHANGE = NEW, write what "CHANGE" equals when dealing with a DECREASE, or an INCREASE

Increase: (1 + pct increase/100)

Decrease: (1 - pct increase/100)

What is the compound interest formula?

Solve using the formula:

"$5,000 invested for 1 year at a rate of 8% compounded quarterly will earn approximately what interest ?"

[image]

where P = principal, r = rate (decimal), n = number of times per year and t = number of years

...

[image]therefore answer = $412 interest

1/100 --> Decimal?

1/50 --> Decimal?

5/8 --> Decimal?

2/3 --> Decimal?

1/9 --> Decimal?

1/10 --> Decimal?

1/20 --> Decimal?

1/8 --> Decimal?

5/4 --> Decimal?

4/3 --> Decimal?

3/2 --> Decimal?

7/4 --> Decimal?

These are the 'core' fractions from which you can derive higher values of the same fraction...

i.e. 1/8 =  0.125 and therefore 5/8 = (0.125 * 5) = 0.625

1/9 = 0.111111rep. = 0.1rep (thus 4/9 = 0.4rep)

1/8 = 0.125

1/7 = 0.143 (to 3dp)

1/6 = 0.16666666 (6 rep.)

1/5 = 0.20

1/4 = 0.25

1/3 = 0.3333333 (3 rep.)

1/2 = 0.5

When should you use fractions and when should you use decimals ?

Fractions  

* Use to cancel factors.

* Also fractions are the best way of exactly expressing proportions that don't have clean decimal equivalents such as 1/7. 

* In some cases it might be easier to compare a bunch of fractions by giving them all a common denominator rather than converting them all to decimals or percents.

Decimals/Percents

* Use to estimate or compare quantities - the implied denominator is 100 so you can easily compare percents (of the same whole) to each other.

Translate the following from 'problem' expressions to their actual meaning... i.e. X percent --> X / 100 :

'of' --> ?

'of Z' --> ?

'Y is X percent of  Z' --> ?

'Y is X percent of  Z' --> alternative way ?

'A is 1/6 of B' --> ?

'C is 20% of D' --> ?

'E is 10% greater than F' --> ?

'G is 30% less than H' --> ?

'The dress cost $J.

Then, it was marked up and sold' --> ?

'What is the profit when the dress cost $J.
Then, it was marked up 25% and sold?' --> ?

'of' --> Multiply

'of Z' --> Z is the whole

'Y is X percent of  Z' --> Y is the Part, Z is the whole... Y = (X/100)Z

Part = (PCT/100)*Whole

'Y is X percent of  Z' --> alternative way -->

Y/Z = X/100 i.e. Part/Whole = PCT%/100 ?

'A is 1/6 of B' --> A = (1/6)B

'C is 20% of D' --> 0.20(D)

'E is 10% greater than F' --> E = (1.10)F

'G is 30% less than H' --> ? 

G = (100% - 30%)H = (0.70)H

'The dress cost $J. Then, it was marked up and sold' --> Profit = Revenue - Cost

'What is the profit when the dress cost $J. Then, it was marked up 25% and sold ?' --> (1.25)J - J...therefore profit = 0.25J

What number is 180% greater than 50?

180/100 = x/50

therefore x = 90

alternatively, 100% of 50 = 50... 20% of 50 = 10. Therefore 80% of 50 = 40 (20%*4=80%).

Therefore, 180% of 50 = 50+40=90.

SO... a number 180% GREATER than 50 = 180% of 50 + 50 = 140.

Alternatively, a number 180% greater than 50 is 280% of 50 = 140.

How do you find out, easily, if one fraction is bigger than another?

Cross-multiply... e.g.

[image]

Explain how the Last Digit Shortcut works, using this example:

(7²)(3³)(9²)

To find the units digit of a product, or a sum of integers, ONLY pay attention to the units digit of the numbers you're working with. Drop any other digits.

This shortcut works because only units digits contribute to the units digit of the product.

e.g. (7²)(3³)(9²): 

Step 1: 7x7 = 49

Step 2: 9x9 = 81

Step 3: 3x3x3 = 27

Step 4 (final step): 9x1x7 = 63

When would you use the Heavy Division Shortcut, and how do you do it?

If the answer is not precise enough, what should you do?

Use the Heavy Division Shortcut when you need an approximate answer to a division problem using decimals that looks complex. 

~ Get a SINGLE DIGIT to the left of the decimal in the denominator. Do this by moving the decimals in the numerator & denominator the SAME DIRECTION and round to whole numbers.

~ Focus on the whole number parts of the numerator and denominator and solve.

IF the answer's not precise enough, keep one (or 2) more decimal places and do long division: e.g. for 1,530,794 / 314,900, instead of doing 15/3, do 153/31 for more accuracy.

Rephrase:

6,782.01 x 10^-3

53.0447 / 10^-2

6,782.01 x 10^{-3 =} 6,782.01 x 1/1000 = 6,782.01 / 1000 = 6.78201

53.0447 / 10^{-2 =} 53.0447 / (1/100) = 53.0447 x 100/1 = 5304.47

Explain the concept of trading decimal places and how it works: e.g. 0.0003 x 40,000

Trading decimal places refers to moving the decimals in the opposite direction the same number of places, when multiplying a very large number and a very small number.  

The reason this technique works is that you're multiplying, and then dividing, by the same power of ten. i.e., you're trading decimal places in one number for decimal places in another.

e.g. 0.0003 x 40,000 = (3 x 10^-4) x (4 x 10⁴) = 3/10,000 x (4 x 10,000) = 3 x 4 = 12

How do you take a power or a root of a decimal?

Give some examples of each...

Also, what shortcuts can be deduced from this?

Split the decimal into 2 parts: an integer, and a power of ten...

e.g. (0.5)⁴ = (5x10^-1)⁴ = 5⁴x10^-4 = 625 x 10^-4 = 0.0625

e.g. ³√0.000027 = (27x10^-6)^1/3 = 27^1/3x10^-2 = 3x10^-2 = 0.03

You can take a shortcut by counting decimal places. For example, the number of decimal places in the result of a cubed decimal is 3 times the number of decimal places in the original decimal. ALSO, the number of decimal places in a cube root is 1/3 the number of decimal places in the original decimal...

What's the reciprocal of √6 , and why?

√6 / 6

because the product of a number and its reciprocal is always 1, multiplying √6 by √6 / 6 = 6/6 = 1 and therefore the fraction above is the reciprocal of √6.

What's a quick way of simplifying a fraction such as this:

1/2

------

3/4

?

Multiply the numerator and denominator by a common denominator in the 'sub fractions'... 

e.g. 1/2 x 4

      --------

       3/4 x 4

= 2/3

Decreasing the DENOMINATOR of a fraction INCREASES/DECREASES the value?

Give an example.

INCREASES the value.

When fraction problems on the GMAT include unspecified numerical amounts (often described by variables), what should you do?

Pick SMART NUMBERS to stand in for the variables. These must be equal to common multiples of the denominators of the fractions in the problem.

DRILL: INCREASING/DECREASING Numerators & Denominators

--------------------------------------------------------

Decide whether the given operation will yield an INCREASE, DECREASE or a result that will STAY THE SAME.  

1. Multiply the numerator of a positive, proper fraction by 1/2

2. Add 1 to the numerator of a positive, proper fraction and subtract 1 from its denominator

3. Multiply both the numerator and denominator of a positive, proper fraction by 3 1/2 

4. Multiply a positive, proper fraction by 3/8

5. Divide a positive, proper fraction by 3/13

6. Add 1/2 to the numerator and denominator of a positive proper fraction

7. Multiply both the numerator and denominator of a positive proper fraction by 5 

7. Add 4 to the numerator and denominator of a positive improper fraction

8. Add 10 to the numerator of a positive proper fraction

9. Add 20 to the numerator of a positive IMPROPER fraction. 

10.

11.

12.

13.

14.

15.

16.

What happens to a positive proper fraction when you:

1. INCREASE the numerator only

2. DECREASE the denominator only

3. DECREASE the numerator only

4. INCREASE the denominator only

5. Add the SAME NUMBER to both the numerator and denominator

What happens to a positive IMPROPER fraction when you:

1. INCREASE the numerator only.

2. DECREASE the denominator only.

3. INCREASE the denominator only.

4. INCREASE the denominator so that it is greater than or equal to the numerator.

5. DECREASE the numerator only.

5. DECREASE the numerator so that it's less than the denominator

A) What is 1,530,794 / (31.49 x 10⁴) to the nearest whole number? (hint: use Heavy Division Shortcut)

B) To get a more precise answer, how would you do it?

A) 1) Eliminate the powers of 10 -->

1,530,794 / 314,900

2) Get a single digit to the left of the decimal in the denominator by dividing the top and bottom by the same power of 10 (i.e. 100,000)

3) 15.30794 / 3.14900 ~= 15/4 = 5

B) Answer: Keep 1 more d.p. and do long division (e.g. 153 / 31 =~ 4.9)