Arithmetic

converting decimals to fractions

1. Find the place value of the last digit of the decimal 2. Write a fraction with that place value as the denominator ex. .35 = 35/100 = 7/20 ex. .004 = 4/1000 = 1/250

fraction properties

1/a/b = b/a (ex. 1/2/3 = 3/2) a/b x b/a = 1 abc/def = a/d x b/e x c/f (ex. 6 x 15 x 24 / 6 x 3 x 4 = 6/6 x 15/3 x 24/4 = 1 x 5 x 5 = 30) --> we can also rearrange the terms to make easier fractions ; all of this does not apply to addition, just multiplication a+b/c = a/c + b/c or a-b/c = a/c - b/c or a+b/c+d = a/c+d + b/c+d (ex. 6-2/11 = 6/11 - 2/11 = 4/11) --> never split the denominator! a/b x b = a (ex. 3/4 x 4 = 3) If a/b = c/d then If we increase the numerator and denominator by the same amount, the fraction is closer to approaching 1 (ex. 2/7 --> 2+10/7+10 = 12/17) Cross simplify before multiplying is they're already simplified on their own (ex. 10/27 x 9/25 = 10÷5/27÷9 x 9÷9/25÷5) = 2/3 x 1/5 = 2/15) To divide fractions, multiply by the reciprocal of the divisor (ex. 1/4÷2/3 = 1/4 x 3/2 = 3/8) (ex. 10/33 ÷ 25/11 = 10/33 x 11/25 = 2/3 x 1/5 = 2/15)

Compound interest

Final = P(1+[r/c])∧(nc) P= principal investment R = annual interest rate as a decimal C = # of compoundings/years n = # of years ex. Amy invests $500 at an annual interest rate of 8% compounded quarterly. What is the value of Amy's investment after 5 years? P = 500 r = .08 c = 4 n = 5 500(1+[.08/4])⁵ ⁴ 500(1=.02)²⁰ For short time periods, and if they ask for a real value, it may be faster to calculate by hand without a formula

percent increases and decreases

We can use 2 eqautions: 1. % Change = change/original value --> rewrite as % (ex. 12/40 = 3/10 = 30/100 = 30%) 2. % Change = change x 100/original value (ex. 1200/40 = 120/4 = 60/2 = 30/1 = 30%) We use this when given 2 #'s and asked about the % increase or decrease

word problems with fractions

"Of" usually means to multiple (ex. 3/10 of 12,000 people plays golf --> 3/10 x 12,000 = 36,000/10 = 3,600 people play golf)

fraction values to memorize

1/5 = .2 = 20% 1/6 = ∼.166 = 16.6% 1//7 = ∼.14 = 14% 1/8 = .125 = 12.5% 1/9 = ∼.11 = 11% We can use these to figure out other fractions; for example, 3/5 = (.2)(3) = .6 and 2/7 = (.14)(2) = ∼.28

successive changes

If price increases by 20%, new price is 100% of original plus 20% of original aka 120% (ex. $30 --> 20% = $6 --> new price is $36) New price = (1± [percent change/100]) x original ex. population was 60 but has increased by 300%, so current population is (1+300/100) x 60 = (1+3) x 60 = 4 x 60 = 240 Use this when given % increase or decrease and asked for a new value; do not use with "of the," but use with "greater than" and "less than" Ex. In 1980, a widget cost 50 dollars. Today, the cost of a widget is 80% of the cost in 1980. --> Use previous equation: 80/100 = x/50 = $40 Ex. In 1980, a widget cost 50 dollars. Today, the cost of a widget is 80% greater than the cost in 1980. Use this equation: (1+[80/100]) x 50 = (1+.08)x50 = 1.08x50 = $90

Simple interest

Interest = (principal)(rate)(time) Principal = initial deposit Rate = decimal form Don't forget to add to initial deposit if it asks for total $

solving percent questions

part/whole = percent/100 (ex. 15 percent of what number is 60?) 1. Label parts (ex. 15 = percent, 60 = part) 2. Solve equation (ex. 60/x = 15/100 = 60/x = 3/20 --> x = 400) ^^Can be used to solve any percent question! An alternative is to calculate percents in our head using 50%, 10%, and 1% 10% = move decimal 1 spot to the left 1% = move decimal 2 spaces to the left 50% = half ex. What is 15% of 62? --> 10% = 6.2 so 5% is 3.1 --> 6.2 + 3.1 = 9.3