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# ACT Scrapbook 16-30

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FC: ACT Hints 16-30 | AnnaMichelle Moore November 24th, 2012 1st period

1: pages 2&3 The only function questions pages 4&5 Midpoint & Distance Formulas pages 6&7 Make it Real pages 8&9 Perimeter, Area, Volume pages 10&11 Donuts pages 12&13 Ratios, Proportions &Cross- Multiplying pages 14&15 Intimidation &Easy Matrices pages 16&17 Art Class pages 18&19 The Mighty Pythagoras pages 20&21 Special Right Triangles pages 22&23 Dr Evil and Mini- Me pages 24&25 Translation pages 26&27 Arithmetic Word Problems pages 28&29 Just Do it- Springboard pages 30&31 The Laws of Exponents | Table Of Contents

2: The Only Function Questions on the ACT | Functions are just a type of equation, like y= mx+b, but to show that an equation is a function, sometimes people replace the "y" with "f(x)" or "g(x)". So just remember f(x) is just a fancy way of saying "y". So f(x)= 2x-1 means the same as y=2x-1

3: f(3) means "plug 3 in for x" f(m) means "plug m in for x" f(g(m)) means "plug g(m) in for x" | Example: If f(x)= negative two x squared minus 3, what is f(-3)? Just plug in -3. Answer: -21

4: Midpoint and Distance Formulas | To find the midpoint of two points, use the formula: Midpoint= To find the distance between two points use the formula: Distance=

5: Example: A line in the standard (x,y) coordinate plane contains the points P(3,-5) and Q (-7,9). What point is the midpoint of segment PQ? | Answer: (-2,2) To find this, you have to find the halfway between the two points (the average). Use the midpoint formula- (x one plus x two divided by 2),(y one plus y two divided by 2)

6: Make It Real | Some questions can be way too theoretical, so it is easiest to take a problem and replace the variables with real numbers. Use numbers that fight the equation. If the numbers in front of the variables are 2 and 3 replace the variables with single digit real numbers also. But be careful with choosing numbers with 0, 1, and 2.

7: When a question with variables is way too theoretical, just "Make It Real." | Solution: Say b=3 and e=4 and solve the equation, c should equal -16 Now, Say b= 5 and e=6. Solve the equation with these variables and you should get -24. So by increasing the two numbers by 2, your c value decreases by 8. | Example: If c= -2b-2(2e-b), what happens to the value of c when b and c are both increased by 2.

8: Perimeter, Area, Volume | Example: A square lot, with area 625 square feet, is completely fenced. What is the length, in feet, of the fence?

9: Memorize basic perimeter area, and volume formulas. To use, plug in what you know, and solve for the variable. | Solution: (s)(s)= 625 s=25 to find Perimeter (4)(25)=100

10: Donuts | To find the area of a shaded region, just subtract the area of the smaller figure from the area of the larger figure. | Solution: Larger circle= 36 pie Smaller circle= 16 pie So 36 pie minus 16 pie is 20 pie and J is the answer

11: The area of a donut equals the area of the big guy minus the area of the donut hole

12: Ratios, Proportions, and Cross-Multiplying | A ratio simply expresses a relationship between two numbers, like boys to girls. There are 10 girls to 12 boys. This is a ratio of 10/12, which can also be reduced to 5/6 or 5:6. If two ratios that are equal such as 10:12 and 5:6, they are called proportions, and if one of those 4 numbers in the proportion are unknown such as the 6 for example you'd have 10/12 = 5/x and you'd cross multiply to find the number. Which means: (12)(5)=(12)(x) and divide each side by 12 to get x alone and to find the answer. Anytime the ACT shows you a proportion, cross multiply!

13: 4 boys to 5 girls could also be expressed as 5 girls to 9 students. A ratio can be a reduced version of the real numbers. And when you see a proportion on the ACT, cross- multiply

14: Intimidation and Easy Matrices | The first is to add matrices. To add matrices, you just add the numbers in each location. The second is to multiply matrices. When they ask you this it is important to remember that the product matrices will have as many rows as the first set and as many columns as the second matrices. For example: a 2 row by 3 column matrix times a 3 row by 4 column matrix will have 2 rows and 4 columns. And the middle numbers must match or the matrices can't be multiplied

15: Relax when you see a matrix question; just treat the matrix like a normal chart. To add matrices, add corresponding numbers; to multiply, remember that the result will have as many rows as the first and as many columns as the second matrix being multiplied; and for any other operation, just follow the instructions that the question provides.

16: Art Class | "Use the Diagram" to estimate an answer. When a diagram is not drawn to scale, redraw it. And when a picture is described, but not shown, draw it! When estimating an answer, translate fractions, square roots, or pi into decimals.

17: Solution: By drawing the diagram depicted in the above question, it is obvious that CE is the longest segment

18: The Mighty Pythagoras | Pythagoreans theorem is very common on every ACT. If you aren't sure what's a leg or where everything goes on the triangle, draw the triangle and label it.

19: When you see a right triangle, try a squared + b squared = c squared

20: Special Right Triangles | Know the rules and what to do when you see a special right triangle!

21: When you see a 30 degree, 45 degree, 60 degree angle in a right triangle, try using the special right triangles.

22: Dr. Evil and Mini-Me | Similar triangles are two triangles where one is a shrinky version of the other, like Dr. Evil and Mini-me. Since one is a shrunken version of the other, all sides are proportional, shortest to shortest and longest to longest.

23: Similar triangles have sides that are proportional. | In the picture above the bigger triangle is twice the size of the smaller. You can tell that by looking at the sides. Side a on the larger triangle is 8 and side a on the little triangle is 4. 8 divided by 4 is 2, which means all the other sides will be 2 times smaller on the little triangle.

24: "Is" Means Equals- Translation | What number means "x" 37 percent means 0.37 "of" and "product" means multiply "less" means minus "sum" or "more than" mean plus "quotient" means divide "is" means equals "is to" or "per" means divide (ratio) and often a proportion "twice a number" means 2x "x increased by 25%" means x+.025x

25: Translate word problems from English to math. | Example: If 5 percent of 20 percent of a number is 24 less than one- quarter of the number, what is the number? | Solution: You would translate all the words into math. One- quarter means 0.25. You'd multiply (0.01)(0.20). Then Add 24 to both sides, and subtract 0.01n from both sides. Divide both sides by 0.24 and you should get 100 as your answer.

26: Arithmetic Word No- Problems | These are very easy even though must freak when they see word problems, but all you need to do is translate the words into math. Translate fractions into decimals and it makes it a lot easier, and any word problems that contain imagery, draw out what's being described.

27: Word problems are no-problem; translate English to math, and translate fractions to decimals. | Solution: first translate the fractions into decimals (1/3= .333 and 2/5= .4) Combine means to add together! So .333+ .4= .733. Now convert all the fraction answers into decimals and find the closet answer to .7333

28: Just Do It!- Springboard | When you see x2-y2, factor it to (x-y)(x+y) Note: x2-y2= (x-y)(x+y) is the ACT's favorite kind of factoring, memorize it! When you see (x-3)(x+2), FOIL it to get x2-x-6 When you see 12/16, reduce it to 3/4 When you see square root of 50, simplify it to 5 to the square root of 2 or just 7.07. When you see 7/8, use your calculator (top divided by bottom) to convert it to 0.875 to convert

29: When something can be factored, foiled, reduced, or simplified- do it. When you have two equations in a question, ask how they relate. Convert fractions to decimals. | Example: If x squared minus y squared= 84 and x-y=6, what is the value of x+y? | Solution: When you see x squared - y squared, factor it! Factor x squared- y squared to (x-y)(x+y) and substitute in 6 for the (x-y) since it equals 6. So 6(x+y)=84. divide 84 by 6 and (x+y)=14. So the answer is 14.

30: The Laws of Exponents I | When multiplying matching bases, add exponents. When dividing matching bases, subtract exponents. Power to a power, multiply exponents Any base to the 0 power equals 1 | Exponents can be used to measure the size of earthquakes and can help represent larger distances.

31: Memorize the laws of exponents.

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