Math 4 Review for Quarter 1 Cumulative Test Name: I. Unit Conversion Units are important in describing the world around us To convert between units: o Method 1: Multiplication/Division Converting to a larger unit divide Converting to a smaller unit multiply o Method 2: Proportions Set- up a proportion (equation like = ) making sure the units line up Cross- multiply and solve for the unknown All conversion factors (like 12in/1ft) equal 1 because they are fractions with the same quantity on top and bottom Multiplicative identity (1) any quantity multiplied by 1 is still the same quantity To make multiple conversions easier, use dimensional analysis: o Multiply by several conversion factors making sure the units will cancel to give you the units you want Conversion Factors: 35 miles 1 hour 5280 feet 1 mile 1 hour 60 mins 1 min = 51.3 ft/sec 60 sec 3 feet (ft) = 1 yard (yd) 5280 feet (ft) = 1 mile (mi) 3.28 feet (ft) = 1 meter (m) 1000 meter (m) = 1 kilometer (km) 12 inches (in) = 1 foot (ft) 2000 pounds (lb) = 1 ton (T) 16 ounces (oz) = 1 pound (lb) 2.2 pounds (lb) = 1 kilogram (kg) 365 days = 1 year Practice Problems 1. Gold costs $500 per ounce. A solid gold nugget is 23.5 ounces. How much is it worth? 2. How many ounces are there in 2.5 tons?
3 feet (ft) = 1 yard (yd) 5280 feet (ft) = 1 mile (mi) 3.28 feet (ft) = 1 meter (m) 1000 meter (m) = 1 kilometer (km) 12 inches (in) = 1 foot (ft) 2000 pounds (lb) = 1 ton (T) 16 ounces (oz) = 1 pound (lb) 2.2 pounds (lb) = 1 kilogram (kg) 365 days = 1 year 3. If a snail travels 8 inches per minute, how fast does it travel in miles per hour? 4. How many inches are there in 110 yards? 5. Sarah can study 20 pages per night. Each page has three homework problems. If she studies 3 nights, how many homework problems will she do? 6. A football field is 100 yards. How long is it in kilometers? 7. A species of fast- growing tree grows at the rate of 2.2 inches per month. How many feet does it grow per year?
II. Order of Operations Rules with Numbers and Polynomials. Order of operations: PEMDAS o Remember: Absolute value bars count as parentheses 4 Fraction bars imply parenthesis: means 4 ( 2 5 1) 2 5 1 Multiplication and division happen together, from left to right Addition and subtraction happen together, from left to right When adding and subtracting polynomials, you can only combine like terms (same combination of variables and exponents) When multiplying polynomials, use a multiplication box to make sure you multiply every term in one polynomial by every term in the other Practice Problems 8. Simplify the following expressions without a calculator a. 3 2 + 16 4 3 b. 3 8 3 2 + 4(5 2) c. 2 + 3 4 3 2 d. " + 2 4 e. 4x + 2 2x + 5 + (x + 4) f. 7x 3x + 1 (x + x 2) g. 7x + 4 (2x 4) h. 3x 1 (x + x 2)
9. Combine like terms, if possible, and then evaluate the expressions for the values given. a. 3a 2b + b(6 b) Evaluate for a = 4 and b = 2 b. 4a + 3 2y 5a 7 + 4y Evaluate for a = 2 and y = 2 c. 2x y 3xy + 4xy 3x y Evaluate for x = 1 and y = 3 III. Manipulating Equations Solving for one variable (like x) in terms of the other variables means you still get x by itself but it will be equal to something involving the other variables, rather than a number Solving for m in F = ma gives m = F a Practice Problems 10. Solve for h: s = 2πr + 2πrh 11. Solve for x: y = x 5 12. Solve for V: W = p(v L)
13. a. The area of a trapezoid is given by the formula A = h B + b, where h is the height of the trapezoid and B and b are the lengths of the two bases. Solve for h. b. Use your formula from part a to find the height of a trapezoid that has an area of 40 in and bases with lengths of 6 inches and 4 inches. 14. a. The volume of a pyramid with a square base is given by the formula: V = s h where s is the length of a side of the base and h is the height of the pyramid. Solve for s. b. Using your formula from part a, find the side length of the base if the volume is 12in 3 and the height is 12 in. IV. Rules of Exponents Exponent rules: a m a n = a m+n a m a = ( n am n a m ) n = a m n a m = 1 1 a m a = m am a 0 =1 a x b x = (ab) x a x b = a x $ # & x " b % a 1 n = n a a m n = n ( a ) m = n a m When polynomials are raised to exponents, watch out for addition and subtraction in the parentheses. You can t distribute the exponent when there is addition or subtraction Do out the multiplication with a multiplication box. (x + 3) 2 x 2 + 9 (x + 3) 2 = (x + 3)(x + 3) = x 2 + 6x + 9
Practice Problems 15. Simplify the following expressions. Your answer should have no negative exponents. a. () b. () c. () d. a b e. 3x 4x 2x f. p q p q g. (x y ) h. (3x y) i. ( 2x ) j. " k. l. "
m. n. 16x y o. (z ) p. " q. 8x y 16. Simplify the following polynomials expressions: a. (x 2) b. 2x + 1 c. 2x 2(x 3) d. x + 1 (2x + 5)
V. Working With Fractions When adding fractions, you must first get the fractions to have the same denominator. 3 4 + 1 3 = 3 3 4 3 + 1 4 3 4 = 9 12 + 4 12 = 13 12 When multiplying fractions, multiply across the tops and multiply across the bottoms. There is no need for the fractions to have the same denominator. 3 4 1 3 = 3 12 = 1 4 When dividing fractions, change it to multiplying by the reciprocal of the second fraction. 3 4 1 3 = 3 4 3 1 = 9 4 Practice Problems 17. Evaluate the following expressions without a calculator. Reduce your final answer, if possible. a. + b. c. + d. e. + f.
VI. Rational Expressions Rational expressions are fractions where there are polynomials in the numerator (top) and/or denominator (bottom) Rational expressions work mostly just like regular fractions work: o Multiply multiply across the tops and multiply across the bottoms o Divide multiply by the reciprocal instead (flip the second fraction) o Add/Subtract must get a common denominator first, then add/subtract across the top When simplifying rational expressions, you don t need to multiply out the bottom but you should multiply out the top You can also simplify rational expressions by cancelling common factors (but make sure there is not addition or subtraction in your way) 2 2x 8 = /2 /2(x 4) = 1 the 2s cancel (x 4) x + 4 x +1 4 the xs can t cancel here because of the addition 1 Practice Problems 18. Simplify the following expressions. a. " b. c. " d. e. f. +
g. h. + i. () j. " "" k. + l. +