10th Grade SAT Math Diagnostic Assessment

Isolate \(10a + 3b\) by multiplying both sides by \((10a + 3b)\), resulting in \(m = q(10a + 3b)\). Dividing by \(q\) gives \(10a + 3b = \frac{m}{q}\).

To solve \( 2x + 3 < 7x - 5 \), rearrange the terms to isolate x: \( 3 + 5 < 7x - 2x \) leading to \( 8 < 5x \) or \( x > \frac{8}{5} \).

Choice D is correct. It’s given that \(5 x = 20\). Multiplying both sides of this equation by \(3\) yields \(15 x = 60\). Therefore, the value of \(15 x\) is \(60\). Choice A is incorrect and may result from conceptual errors. Choice B is incorrect and may result from conceptual errors. Choice C is incorrect and may result from conceptual errors.

Choice B is correct. It’s given that the shop’s inventory starts with \(4,500\) paper cups and that the manager estimates that \(70\) of these paper cups are used each day. Let \(x\) represent the number of days in which the estimated supply of paper cups will reach \(1,700\). The equation \(4,500 -70 x = 1,700\) represents this situation. Subtracting \(4,500\) from both sides of this equation yields \(-70 x = -2,800\). Dividing both sides of this equation by \(-70\) yields \(x = 40\). Therefore, based on this estimate, the supply of paper cups will reach \(1,700\) in \(40\) days. Choice A is incorrect. After \(20\) days, the estimated supply of paper cups would be \(4,500 -70(20)\), or \(3,100\) cups, not \(1,700\) cups. Choice C is incorrect. After \(60\) days, the estimated supply of paper cups would be \(4,500 -70(60)\), or \(300\) cups, not \(1,700\) cups. Choice D is incorrect. After \(80\) days, the estimated supply of paper cups would be \(4,500 -70(80)\), or \(-1,100\) cups, which isn't possible.

Choice C is correct. Each term in the given expression, \(12 x + 27\), has a common factor of \(3\). Therefore, the expression can be rewritten as \(3(4 x)+ 3(9)\), or \(3(4 x + 9)\). Thus, the expression \(3(4 x + 9)\) is equivalent to the given expression. Choice A is incorrect. This expression is equivalent to \(108 x + 12\), not \(12 x + 27\). Choice B is incorrect. This expression is equivalent to \(324 x + 27\), not \(12 x + 27\). Choice D is incorrect. This expression is equivalent to \(27 x + 72\), not \(12 x + 27\).

Choice C is correct. According to the line of best fit, a planetoid with a distance from the Sun of 1.2 AU has a predicted density between and . The only choice in this range is 4.6.Choices A, B, and D are incorrect and may result from misreading the information in the scatterplot.

The correct answer is \(88\). It's given that there are a total of \(275\) attendees and each attendee is assigned to either group A, group B, or group C. It's also given that if one of these attendees is selected at random, the probability of selecting an attendee who is assigned to group A is \(0.44\) and the probability of selecting an attendee who is assigned to group B is \(0.24\). It follows that there are \(0.44(275)\), or \(121\), attendees who are assigned to group A and \(0.24(275)\), or \(66\), attendees who are assigned to group B. The number of attendees who are assigned to group C is the number of attendees who are not assigned to group A or group B. In other words, the number of attendees who are assigned to group C is the total number of attendees minus the number of attendees who are assigned to group A and group B. Therefore, the number of attendees who are assigned to group C is \(275 -121 -66\), or \(88\).

Choice B is correct. It′s given that the airplane descends at a constant rate of \(400 feet per minute\). Since the altitude decreases by a constant amount during each fixed time period, the relationship between the airplane′s altitude and time is linear. Since the airplane descends from an altitude of \(9,500 feet\) to \(5,000 feet\), the airplane′s altitude is decreasing with time. Thus, the relationship is best modeled by a decreasing linear function. Choice A is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Choice C is correct. In order to use a sample mean to estimate the mean for a population, the sample must be representative of the population (for example, a simple random sample). In this case, Tabitha surveyed 20 families in a playground. Families in the playground are more likely to have children than other households in the community. Therefore, the sample isn’t representative of the population. Hence, the sampling method is flawed and may produce a biased estimate.Choices A and D are incorrect because they incorrectly assume the sampling method is unbiased. Choice B is incorrect because a sample of size 20 could be large enough to make an estimate if the sample had been representative of all the families in the community.

Choice C is correct. The y-intercept of a graph is the point where the graph intersects the y-axis. The line graphed intersects the y-axis at the point \((0, 5)\). Therefore, the y-intercept of the line graphed is \((0, 5)\). Choice A is incorrect and may result from conceptual errors. Choice B is incorrect and may result from conceptual errors. Choice D is incorrect and may result from conceptual errors.

Choice D is correct. In the figure shown, the angles with measures \(w °\) and \(z °\) are vertical angles. Since vertical angles are congruent, \(w = z\). Therefore, if \(w = 136\), the value of \(z\) is \(136\). Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect. This is the measure, in °s, of an angle that's supplementary, not congruent, to the angle with measure \(w °\). Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is correct. It's given that the number of coins in the collection increased from \(9\) to \(90\). It follows that the number of coins in the collection increased by \(90 -9\), or \(81\). Let \(x percent sign\) represent the percentage that \(81\) is of \(9\). The value of \(x\) can be found using the proportion \(81/9 = x/100\), or \(9 = x/100\). Multiplying both sides of this equation by \(100\) yields \(900 = x\). Thus, when the number of coins in the collection increased from \(9\) to \(90\), the percent increase was \(900 percent sign\). Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors.

Choice C is correct. If the values in a data set are ordered from least to greatest, the median of the data set will be the middle value. Since each data set in the choices is ordered and contains exactly 9 data values, the 5th value in each is the median. It follows that the median of the data set in choice C is 32. The sum of the positive differences between 32 and each of the values that are less than 32 is significantly smaller than the sum of the positive differences between 32 and each of the values that are greater than 32. If 32 were the mean, these sums would have been equal to each other. Therefore, the mean of this data set must be greater than 32. This can also be confirmed by calculating the mean as the sum of the values divided by the number of values in the data set: .Choices A and B are incorrect. Each of the data sets in these choices is symmetric with respect to its median, so the mean and the median for each of these choices are equivalent. Choice D is incorrect. The median of this data set is 207. Since the sum of the positive differences between 207 and each of the values less than 207 is greater than the sum of the positive differences between 207 and each value greater than 207 in this data set, the mean must be less than the median.

The correct answer is 30. In the figure given, since is parallel to and both segments are intersected by, then angle BDC and angle AEC are corresponding angles and therefore congruent. Angle BCD and angle ACE are also congruent because they are the same angle. Triangle BCD and triangle ACE are similar because if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. Since triangle BCD and triangle ACE are similar, their corresponding sides are proportional. So in triangle BCD and triangle ACE, corresponds to and corresponds to . Therefore, . Since triangle BCD is a right triangle, the Pythagorean theorem can be used to give the value of CD: . Taking the square root of each side gives . Substituting the values in the proportion yields . Multiplying each side by CE, and then multiplying by yields . Therefore, the length of is 30.

The correct answer is \(11/28\). The cosine of an acute angle in a right triangle is defined as the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse. In the triangle shown, the length of the leg adjacent to the angle with measure \(x °\) is \(11\) units and the length of the hypotenuse is \(28\) units. Therefore, the value of \(cosine x °\) is \(11/28\). Note that 11/28, .3928, .3929, 0.392, and 0.393 are examples of ways to enter a correct answer.