MATH. Part 1. INTEGERS. ADDITION - Same sign : add the numbers and copy the sign - Opposite signs : subtract the numbers and copy the sign of the larger number MULTIPLICATION/ DIVISION - Same sign : positive (+) - Opposite signs : negative (-). Order of Operations. GEMDAS
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- Same sign: add the numbers and copy the sign
- Opposite signs: subtract the numbers and copy the sign of the larger number
- Same sign: positive (+)
- Opposite signs: negative (-)
Groups, Exponent, Multiplication, Division, Addition, Subtraction
Prime and Composite Numbers
- Number greater than 1 whose only divisor is 1 and the number itself
- example: 7 : 1, 7
- Any integer which has other divisors aside from 1 and the number itself
- example: 12: 1, 2, 3, 4, 6, 12
- Similar fractions: Add or subtract the numerators and copy the denominators.
- Dissimilar fractions: Get the Least Common Denominator (LCD) , get the equivalent fractions, add or subtract the numerators.
- Multiply the numerators. Multiply the Denominators.
- Multiply the dividend by the reciprocal of the divisor
- Align the decimal points. Add zeroes so they have the same length. Add or subtract them.
12.34 + 5.6789 = ?
- Multiply the numbers normally. Count the total number of decimal points of both numbers. This will determine the number of decimal places of the product.
4.2 x 1.2 = ?
*2 decimal points
- quantities change in the same manner
-solved using proportionality or by division
How many hours would a 675 mile trip by train take if a 45 mile trip takes 30 minutes?
45 mile : 30 minutes = 675 mile : x hr
30 minutes = ½ hr
45x = ½ (675)
x = 7.5 hr
- quantities change in the opposite manner
- solve by multiplying the first quantity by the second and setting an equation for the different cases
Fifty workers can assemble one lot of an item in one day(24 hr). If 25 more workers were hired, how many more hours would it take to assemble one lot?
50 (24 hr) = (50 + 25)(x hr)
x = 16
- move the decimal point two places to the left
- example: 42% 0.42
- move the decimal point two places to the right
- example: 0.1314 13.14%
- divide the numerator by the denominator
- example: = 0.67
- divide the decimal digits by the last decimal place
- example: 4.5 = 4 = 4
- divide the percent value by 100
- example : 78% =
- divide the numerator by the denominator, the answer should be in decimal form to be changed to percent OR
- multiply the fraction by 100, simplify then add the % sign
= 0.375(100) = 37.5%
*Always change % to decimals or fractions before solving
What is 35% of 720?
x = (720)
x = 252
= %Discount (Original Price)
= Original Price – Discount
= Original Price – (1 - %Discount)
Tom paid only P1020 for a pair of running shoes that was marked 15% OFF. What was the original price?
1020 = x(1 – 0.15)
1020 = 0.85x
= Discount/ Original Price
What was the percent discount if a souvenir item originally marked P75 was bought for only P68.25?
% Discount = 75 – 68.25
% discount = 9 %
Operations on Functions- substitute the variable in the function with the value inside the parenthesis, simplify and then perform the indicated operation
- (f + g)x = f(x) + g(x)
f(x) = 2x3 – 2x2 + 5; g(x) = x2 + x - 6
(f+g)x = (2x3 – 2x2 + 5) + (x2 + x – 6)
(f+g)x = 2x3-x2 + x – 1
- (f - g)x = f(x) - g(x)
f(x) = 4x - 2; g(x) = x + 8
(f - g)x = (4x - 2) - ( x + 8)
(f - g)x = 3x - 10
- (f • g)x = f(x) • g(x)
f(x) = x - 1; g(x) = x
(f • g)x = (x - 1) • (x)
(f • g)x = x2 – x
- ; g(x) ≠ 0
f(x) = x -2; g(x) = 3x - 5
- function within a function
= (f g)x = f(g(x))
f(x) = 2x + 1; g(x) = x2 - 1
(f g)x = 2(x2 – 1) + 1
(f g)x = 2x2 - 1
- DENOMINATOR of a function should not be equal to zero
- RADICAND with an even number as its index should be always be greater than or equal to zero.
Find the domain of the function:
4x -2 ≠ 0
4x ≠ 2
x ≠ ½
Equations involving 1 variable
- Combine similar terms and solve algebraically, using PEMDAS.
3x – 9 = 2x – 6
3x – 2x = -6+9
x = 3
- Use elimination or substitution
2x + 2y = -2
x - y = 1
2x + 2y = -2
2(x – y = 1)
2x + 2y = -2
2x – 2y = 2
4x = 0
x = 0
0 – y = 1
y = -1
x = y + 1
2(y + 1) + 2y = -2
2y + 2 + 2y = -2
4y = -4
y = -1
x = -1 + 1
x = 0
|2x – 4| = 6
2x – 4 = 6 2x = 10 x = 5
2x – 4 = -6 2x = -2 x = -1