EC-6 Core subjects Math: Algebra and Patterns

What percent is represented by the shaded area of the decimal square?

24% EXPLANATION: There are 24 small squares shaded of the 100 total squares; therefore 24% is the correct answer.

A baseball mitt is on sale for 30% off. If the regular price is $78, what is the sale price, excluding tax?

$54.60 EXPLANATION: There are a couple of different ways to work this problem. You can simply take 30% of $78 (0.3 • 78 = 23.4 or $23.40) and find out how much you will save, then subtract the savings from the original price: $ 78 - 23.40 = $ 54.60. You can also obtain the answer by: 100% - 30% = 70% and 0.70•78 = $ 54.60.

James has saved $35.25. He wants to save his money to buy a bicycle that costs $85.00. His brother's bike cost $92.00. If sales tax is 8%, about how much more must he save to afford purchasing his bike, including tax?

$60 EXPLANATION: . One quick measure to find most sales tax amounts is add a tenth to the original price, so on $85.00 tax would come to about $8.50. $85.00 + $8.50 = $93.50. Notice this is an overestimate so James' target will be a bit more than he actually needs. James needs to save about $93. If he has saved about $35, he will need an additional $58. ($93 - $35 = $58) Therefore, if rounded up this would be the best choice: $60. When dealing with money, generally an overestimate is more reasonable.

Standard Deviation

(s) the positive square root of the variance

8oz= ______cup

1 cup

Which of the following lists all the factors of a composite number? 1, 7, 21 1, 11 2, 3, 6 1, 5, 25

1, 5, 25 EXPLANATION: When listing ALL of the factors of a composite number the number itself and 1 must always be included. Remember also that a composite number is a positive integer that has a positive divisor other than one or itself. The factor pairs for the composite number 25 are: 1 and 25; 5 and 5. Notice that 5 x 5 causes the 5 to be a repeated factor. So when listing the factors of the composite number 25 in order, we have: 1, 5, and 25.

There is a 15% increase in tuition at UT for next fall. If the current tuition is $3500 per semester, which equation could be used to find x, the new tuition for the fall? 0.15 * 3500 = x 1.15 * 3500 = x (15/100)= (x/3500) .85 * 3500 = x

1.15 * 3500 = x EXPLANATION:. With a 15% increase, the tuition of 3500 will increase: 100% + 15% = 1 + 0.15 = 1.15. So, 1.15 • 3500 = x would yield the new tuition rate. .85 * 3500 = x

In Anytown ISD, 13 out of every 20 students ride the bus. Which ratio compares the number of students who ride the bus to those who do not? 13: 20 7: 20 7: 13 13:

13: 7 EXPLANATION: This is the equation that compares riders to non-riders. 13 out of 20 ride the bus. This means that the complement of this relationship, those who do not ride the bus, is 20 - 13 = 7. So the ratio of those who ride the bus TO those who do not ride the bus is: 13:7. When writing ratios, remember that order is important. 13:7 is not the same as 7:13 just like the fractions 13/7 ≠ 7/13 (recall that another way to write the ratio 13:7 is 13/7).

The state sales tax is 7.5%. Which number could also represent 7.5%?

3/40 EXPLANATION: There is always the option of arriving at the answer by eliminating incorrect answer choices, but it is always a good idea to double-check the final choice. In the case of this question, 3/40 = 3 ÷ 40 = 0.075 = 75/1000=7.5/100 = 7.5%

The ratio of Antone's height to arm span is the same as the ratio of his brother's height to arm span. About how tall is Antone?

39" EXPLANATION: This is the correct choice, 39", is found by solving the proportion: (56/60)=(36/x). In this proportion, x is Antone's height. In the first ratio, 56 is Antone's brother's arm span and 60 is his height. In the second ratio, 36 is Antone's arm span, and x, his height. Proportions are solved by cross- multiplying. When cross multiplication is performed on this proportion, we get: 56•x = 60•36. When simplified we get: 56x = 2160; then you divide each side by 56 resulting in x = 2160/56=38.57 but because the question says "about" you would use 38.57 ≈ 39".

12π/ 9 is approximately equivalent to:

4 EXPLANATION: 4 is the correct answer. Recall that a close estimation for pi is 3. So 12π = 36 and 36/9 = 4.

If each square in the decimal square has a value of 0.1, then which of the following is the decimal numeric representation of the shaded area?

4.8 EXPLANATION: If each square has a value of 0.1, then each column of 10 squares would be equivalent to 10 • 0.1 = 1. There are 4 columns shaded for a total of 4 and 8 additional small squares for 0.1 • 8 = 0.8. The total value of the shaded area is 4.8.

A sixth grade class is asked to estimate the answer to the following question: 75.8 + 326.79 + 488.92 / 11 = _____. Which of the following would be the best answer?

450 EXPLANATION: This question checks for understanding of order of operations. Since this problem involves both addition and division, division MUST be done first. Correct order of operations is Parenthetical expressions, exponents, multiplication/division (left to right whichever comes first), and finally addition/subtraction (left to right whichever comes first). So the 488.92 / 11 is performed first. Since this is an estimation problem, it would be about 500 / 10, which is 50. Then 75.8 + 326.79 is about 400. Adding the 50 and 400 we get about 450

The Booster Club at Martin MS is selling spirit buttons for homecoming. The buttons cost $0.75 to make and will be sold for $2 each. How many buttons, b, must be sold to make a profit of $500?

500 = $2 b - $0.75b EXPLANATION: The profit is equal to the selling price minus whatever costs are applicable. So if the profit is to be $500 then enough buttons must be sold to reach that profit. If we are selling the buttons for $2 each but it costs $0.75 to make each one, then there is a profit of $2 - .75 or $1.25 on each button. How many buttons will we have to sell to reach $500 profit: $500 = $1.25b. This option is: $500 = $2b - 0.75 c = 1.25 b. So, the Booster Club will have to sell 400 buttons.

Larry started the following number pattern: 900, 888, 876, 864, ... Which number could not be a part of Larry's pattern? 736 816 756 624

736 EXPLANATION: As you look at the sequence of numbers, you should see that the numbers in the sequence decrease by 12. 736 cannot be reached by subtracting 12 any number of times. Subtracting 12 thirteen times will get 744, and fourteen times will result in 732

Which of the following word problems is the correct question to be written and solved with the given equation? 8 - 5 + 3 = 6

8 birds were sitting on a fence. 5 birds flew away and 3 more birds landed on the fence. How many birds are on the fence now?

How many 1/2 ounces are in 8 ounces?

8 divided by half = 16 ounces

Fact Family

A fact family consists three numbers which are used in two addition problems with the addends reversed (a + b = c and b + a = c) and two subtraction problems (c - b = a and c - a = b).

Linear Equation

A linear equation is an equation whose graph is a line.

Mrs. Taylor has 3 ½ yards of material to make spirit flags for the high school football game. Each flag requires ¼ yard strip of material. Which picture above is an accurate picture of this situation and its solution?

C EXPLANATION: This illustrates each yard of the 3 yards divided into 4 parts (1/4 yd) and the ½ yd separated into ¼ yd pieces. There a total of 14 pieces, each equivalent to ¼ yd, so 14 flags can be made.

Jana is playing a game called Guess My Rule. Her friends choose a number (input), Jana subtracts 3 and tells her friends the result (output). The game continues until one of her friends can predict what their output number will be and can correctly state Jana's rule. Which graph correctly shows possible values of input/output for Jana's rule?

B EXPLANATION: If we examine Jana's rule, (input a number, subtract 3, give output), then if we input "0", the output would be 0 - 3 = -3. Only option B has the ordered pair (0,-3) as a point. As a side note, the ability to eliminate choices is an excellent test taking and problem solving strategy. If you can eliminate some of your options, and you have to make a guess, then your odds of getting the correct answer is greatly increased. In this particular case, the odds of getting the correct answer to the original question if you had to arrive at your choice by guessing, would be only 25%. By eliminating two choices, your odds of purely guessing, have increased to 50%.

Maria baked 6 dozen cookies for her classmates. There are 28 students in her class; each child received 2 cookies and Maria gave 6 cookies each to her teacher and her principal. Which equation could be used to find C, the number of cookies she had left over? C = 6 * 12 - 2 * 28 + 2 * 6 C = 6 * 12 - (2 * 28 - 2 * 6) C = 6 * 12 - (2 * 28 + 2 * 6) C = 6 * 12 - 2 * 28 - 6

C = 6 * 12 - (2 * 28 + 2 * 6) EXPLANATION: 6 * 12 gives us the total number of cookies in 6 dozen: 72 cookies. Then if each child receives 2 and there are 28 children, 2 * 28 = 56; if the teacher and the principal each receive 6, that is another 12. So we have 72 - 56 - 12 = 4. Maria will have 4 cookies left.

When considering the addition problem 1/3 + 3/8, which of the following statements is true? I. The LCD = 24 II. 3 and 8 are relatively prime

I and II EXPLANATION: Two numbers are relatively prime if they have no common factors besides 1. Because 3 and 8 are relatively prime, there is no whole number, other than 1, that will divide both 3 and 8 evenly without a remainder. Because 3 and 8 are relatively prime their least common denominator (LCD) can be found using the formula 3 • 8 = 24. Therefore 24 is the LCD.

Which of the following situations might require the use of a common denominator? I. Addition of Fractions II. Multiplication of fractions III. Subtraction of Fractions IV. Division of fractions

I and III EXPLANATION: Unless fractions have like denominators, you must always find a common denominator before you can add or subtract. Finding a common denominator requires finding a common multiple of the two (or more) denominators. It is important to note that you do not have to find the LCM or lowest common denominator, any common multiple will work. However, if the LCM is found and used, there will be considerably less simplifying to do in order to reduce to the lowest terms. Multiplication and division never require finding a common denominator.

I. Y=X II. Y=X2 III. Y=X-2 IV. Y=-2X2+3 V. Y=-3X-5 VI. Y=-1/3X-5/3 Which of the following lists ALL the equations that are linear? I, III, and IV I, II, III, and IV I, II, and IV I, III, V, and VI

I, III, V, and VI EXPLANATION: A linear equation is an equation whose graph is a line. The x and y variables contained in a linear equation are always first degree, meaning that they have no exponents - or they have exponents of 1; which are not usually written. So, x and x^1 are equivalent expressions. Of the equations I - VI, only I, III, V, and VI are linear. Equations II and IV are quadratic equations. All quadratic equations contain one second degree variable: a variable with an exponent of 2. When quadratic equations are plotted, their graphs are parabolas.

Sixteen teachers placed a book order for books to be used in their classrooms. The bill, totaling $350, is to be shared equally. How much will each teacher pay?

III EXPLANATION: This situation involves money. We are looking for an answer that can be "translated" into money. You can use this option by taking the exact answer of 21.875 and rounding it to $21.88.

A field trip to the planetarium is being planned. If each bus will hold 16 people, how many busses will be needed to transport 350 children and teachers? Which answer above is the most appropriate for this situation?

IV EXPLANATION: This is an example of a situation in which an overestimation is required. A bus can carry only 16 people. So 21 busses will not allow for everyone to have a ride and it does not make sense to take a fraction of a bus. Therefore, 22 busses must be taken.

Which situation could best be represented by the equation: 12x = 54? Marty earns $12 for typing a paper. If her rate is $ 54 per hour, what is x, the number of hours it actually took to type the paper? Marty collected 12 dozen eggs every day for 54 days. What is x, the total number of dozens of eggs she collected? Marty had 54 minutes left on his cell phone plan. If he uses 12 minutes, what is x, the number of minutes remaining on his cell phone plan? Marty made car payments on her car for 54 months until it was paid off. What is x, the number of years it took Marty to pay off her car?

Marty made car payments on her car for 54 months until it was paid off. What is x, the number of years it took Marty to pay off her car? EXPLANATION: This is the correct situation for the given equation. 54 months divided by 12 months per year gives us 4 ½ years.

Susan 5+2=7 Bob 3+6=9 Jose 9-4=5 Renee 7-2=5 Raul 7+2=9 Susan's second grade class is studying fact families. Each student is given a fact card and asked to find other members of their family. Above are the cards received by 5 students. Which students have cards that belong to the same fact family?

Susan and Renee EXPLANATION: A fact family consists three numbers which are used in two addition problems with the addends reversed (a + b = c and b + a = c) and two subtraction problems (c - b = a and c - a = b). In this case the digits 2, 5, and 7 and only those digits, must be used in each addition and subtraction fact.

Mean

the mean is found by adding all of the numbers together and dividing that sum by the number of numbers. Mean is also known as the average

Mode

the most frequent number

Quadratic Equations

contain one second degree variable: a variable with an exponent of 2. When quadratic equations are plotted, their graphs are parabolas.

Median

found by first putting all the numbers in order from least to greatest the median can be either the middle number that has an odd amount of numbers or when there is an even set of numbers you find the average of the middle two numbers

Composite number

is a positive integer that has a positive divisor other than one or itself

Variance is found by

squaring the differences and then adding the squares together

Range

take the highest value in the set of numbers and subtract it from the lowest value in the set of numbers

If 15 ml is equivalent to ½ oz, which equation could be used to find x, the number of ml in 1 cup? x = 15 / 1/2 + 8 x = 15 * 1/2 * 8 x = 15 / 8 * ½ x = 15 * 8 / ½

x = 15 * 8 / ½ EXPLANATION: There are 8 oz in one cup, but the problem references ½ oz. So, how many ½ ounces are in 8 oz? 8 ÷ 1/2 = 16. If each 1/2 ounce is equivalent to 15 ml, then 15 • 16 would give us the number of ml in 8 ounces or 1 cup = 240 ml. x = 15 • 8 ÷ ½ = 120 ÷ ½ = 240 ml.

Which expression best represents y in terms of x? y = x - 2 y = 2x - 3 y = 1 - 2x y = -(x + 4)

y = 1 - 2x EXPLANATION: The expression y = 1 - 2x is the only expression that satisfies ALL of the possibilities on the chart. By plugging the value of X, found in the chart, into the equation, the result will equal the value of Y indicated in the chart.