Numbers and Algebra - Patterns and Algebra - Quiz 1
The graph shows the relationship between the number of hours that the Torres' babysitter watches their children and the amount of money that the Torres family owes the babysitter. According to this graph, how many dollars per hour does the Torres family pay their babysitter?
$12.50 The dollars per hour is the rate of change and, therefore, the slope of the line, which can be calculated by finding \frac{change in y \left( \$ \right) }{change in x \left( hours \right) }changeinx(hours)changeiny($). First, identify at least 2 points on the line, for example, (0, 0) and (4, 50). The change in y-values is the change from 0 to 50, which is 50. The change in x-values is the change from 0 to 4, which is 4. So the slope is \frac{\$50}{4 hours}=\frac{\$25}{2 hours}=\frac{\$12.50}{1 hour}=\$12.50/hour4hours$50=2hours$25=1hour$12.50=$12.50/hour.
The scatterplot shows the relationship between sales at an ice cream shop and temperature. Based on the best fit line, shown, which of the following is the best estimate of the sales when the temperature is 21°C?
$450 When the temperature is 21°C (found on the x-axis), the sales are about $450 (the corresponding y-value).
A sweater is on sale for $43.75. It is 30% off the original price. What is the original price?
$62.50 Set up a ratio to solve this problem: 43.75/x = 7/10. Cross multiply so you have 43.75 * 10 = 7x. Divide both sides by 7 to isolate x. So, 437.5/7 = $62.50.
The graph of the function f(x)f(x) is shown. For what value(s) of xx does f(x) = 1f(x)=1?
-4, -1, and 4 The expression "f(x)f(x)" means "the value of the function," which is the y-coordinate of the function for that x. So, "f(x) = 1f(x)=1" means that the y-coordinate equals 11. Therefore, the question is asking for the x-value(s) with a y-coordinate of 11. This question is answered with a simple look from left to right across the function for intersection with the horizontal line y = 1y=1.
What is the y-coordinate of the point on the line drawn when x = 125?
175 First, write an equation that gives the output, y, for any input, x. This is a straight line, and so the equation can be a linear equation in the form y=mx+b. The line passes through the origin, so b=0. The slope can be found using the two points on the graph: (25, 35) and (50, 70). Therefore, m=\frac{rise}{run}=\frac{35}{25}=\frac{7}{5}m=runrise=2535=57. So the equation y=\frac{7}{5}xy=57x represents the line. The question asks for the value of y when x=125, so substitute 125 in for x in the equation to get y: y=\frac{7}{5} \left( 125 \right) =\frac{875}{5}=175y=57(125)=5875=175.
Simplify: 200 - 3(6 - 2)³ + 10
18 To get the correct final value of 18, the Order of Operations must be followed. In this problem, there are several operations occurring: subtraction, multiplication, use of exponents, and addition. Because the order of operations, commonly abbreviated by the acronym PEMDAS, requires that what is inside parentheses be addressed first, the first step in this problem is to perform the subtraction of 6 - 2 = 4. From there, the problem becomes 200 - 3(4)³ + 10. Once the work inside of the parentheses has been completed, exponents should be handled next. In this case, 4 is raised to the third power, resulting in 200 - 3(64) + 10. The subtraction, multiplication, and addition remain. As "M" precedes both "A" and "S" in PEMDAS, multiplication of the 3 with 64 must be completed before subtraction or addition can occur, leaving 200 - 192 + 10. In this next stage, the fact that subtraction is a form of addition (subtraction is addition of opposites), and so addition and subtraction form a pair of equally ranked operations, means that the subtraction and addition in this problem are to be simplified in the order in which they appear when reading the problem from left to right. Accordingly, 200 - 192 + 10 = 8 + 10 = 18.
Simplify: 15 - 3(8 - 6)²
3 To get the correct final value of 3, the Order of Operations must be followed. In this problem, there are several operations occurring: subtraction, multiplication, and use of exponents. Because the order of operations, commonly abbreviated by the acronym PEMDAS, requires that what is inside parentheses be addressed first, the first step in this problem is to perform the subtraction of 8 - 6 = 2. From there, the problem becomes 15 - 3(2)². Once the work inside of the parentheses has been completed, exponents should be handled next. In this case, 2 is raised to the second power, resulting in 15 - 3(4). The subtraction and multiplication remain, but as "M" precedes "S" in PEMDAS, multiplication of the 3 with 4 must be completed before subtraction can occur, leaving 15 - 12 = 3. Note that a common error in this problem may include erroneously distributing the 3 from the front of the parentheses into the quantity "8 - 6" inside the parentheses. Because distributing 3 is a form of multiplication, and because exponents must be used before multiplication can occur, the 3 cannot be distributed to the quantity in the parentheses, but must be multiplied only after the power is raised.
Which of the following word problems is a valid question to prompt writing the equation: 8 - 5 + 3 = n8−5+3=n?
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? The best response begins with 8 items, uses language to lower the initial amount of items by 5, then to raise the resulting amount by 3, and finally asks how many items there are in the end. Only the answer choice with the birds begins with 8 items (8 birds), has 5 removed (5 fly away), 3 added in (3 more birds landed on the fence), and then asks "How many birds are on the fence now?"
What is the 10th term of the geometric sequence 50, 40, 32, 25.6, 20.48,...?
A(10) ≈ 6.71 A(10) ≈ 6.71 comes from seeing that the sequence is shown to have a₁ = 50, r = 40 ÷ 50 = 0.8, and n = 10. The question is answered correctly by keeping an appropriate variable expression in the place of A(n) and inputting all known values so that the formula A(n) = a₁(r)ⁿ⁻¹ becomes A(10) = 50(0.8)¹⁰⁻¹. When the order of operations is followed correctly, exponents will be simplified before any multiplication occurs. The resulting equation becomes A(10) = 50(0.8)⁹ and then A(10) = 50(0.134217728) [or, fractions can be used for r so that r = 4/5 and r⁹ = 262,144/1,953,125 so that the equation is A(10) = 50(262,144/1,953,125)]. The final answer is the product of 50 and the decimal or fraction above, and so A(10) ≈ 6.71 when rounded to the nearest hundredth.
Miss Wilson's class is learning about unit rates. They are comparing costs from the grocery store to see if the store brand is a better deal than the name brand. Douglas and Kaitlyn are volunteering to answer questions during class, but few others are. What should Miss Wilson do?
Provide wait time before calling on someone to answer Providing this wait time allows more students to think through the question and their answer. It is especially helpful if students are told they cannot raise their hand until asked to do so. This gives students additional think time and builds confidence.
