Place Value & Number Lines - Accuplacer Arithmetic

Consider the number 8,142.359. A new number is created by swapping the digit in the hundreds place with the digit in the hundredths place. What is the new number?
Answer: 8,542.319 In the number 8,142.359, the digit in the hundreds place is 1. The digit in the hundredths place is 5. Swapping these two digits results in the new number 8,542.319.
On a number line, three consecutive tick marks represent $1.2$ , $1.25$ , and $1.3$ . A point $P$ is located exactly $\frac{3}{5}$ of the way from the first tick to the third tick. Which comparison correctly describes $P$ 's position relative to $1\frac{1}{4}$ ?
Answer: $P > 1\frac{1}{4}$ because $P = 1.26$ and $1\frac{1}{4} = 1.25$ When you encounter problems about finding positions on a number line, you need to determine the exact location of point P and then convert mixed numbers to decimals for accurate comparison. To find P's position, first identify the total distance from the first tick (1.2) to the third tick (1.3): $1.3 - 1.2 = 0.1$ . Since P is located $\frac{3}{5}$ of the way from 1.2 to 1.3, you need to find $\frac{3}{5}$ of this distance: $\frac{3}{5} \times 0.1 = 0.06$ . Therefore, P's position is $1.2 + 0.06 = 1.26$ . Next, convert the mixed number $1\frac{1}{4}$ to a decimal: $1\frac{1}{4} = 1 + \frac{1}{4} = 1 + 0.25 = 1.25$ . Since $1.26 > 1.25$ , we have $P > 1\frac{1}{4}$ , making choice D correct. Choice A incorrectly calculates P as 1.27 and makes an error converting $1\frac{1}{4}$ to 1.26. Choice B wrongly places P at 1.24, which would require P to be only $\frac{2}{5}$ of the way from first to third tick, not $\frac{3}{5}$ . Choice C suggests P equals $1\frac{1}{4}$ , but this would only be true if P were at the middle tick mark (1.25), which occurs at exactly $\frac{1}{2}$ of the way, not $\frac{3}{5}$ . Remember: When working with fractions of distances on number lines, always calculate the total distance first, then multiply by the given fraction, and finally add to your starting point.
Let $X$ be the number 81.35. Let $Y$ be the number formed by swapping the digit in the tens place of $X$ with the digit in the tenths place of $X$ . What is the sum $X+Y$ ?
Answer: 113.20 In the number $X = 81.35$ , the digit in the tens place is 8, and the digit in the tenths place is 3. Swapping these digits creates the new number $Y = 31.85$ . The sum is $X + Y = 81.35 + 31.85 = 113.20$ .
Which of the following numbers is located between $\frac{4}{5}$ and $\frac{7}{8}$ on a number line?
Answer: 0.810 First, convert the fractions to decimals. $\frac{4}{5} = 0.80$ and $\frac{7}{8} = 0.875$ . The question asks for a number between 0.80 and 0.875. The only option in this range is 0.810.
What is the sum of the number 4.719 and the value of the digit in its hundredths place?
Answer: 4.729 The number is 4.719. The digit in the hundredths place is 1, and its value is 0.01. The sum is $4.719 + 0.01 = 4.729$ .
Two numbers $x$ and $y$ satisfy $4.7 < x < 4.8$ and $4.75 < y < 4.85$ . Based on place value reasoning, which statement about their positions on a number line is necessarily true?
Answer: $x$ and $y$ must overlap in the interval $(4.75, 4.8)$ The interval for x is (4.7, 4.8) and for y is (4.75, 4.85). These intervals intersect in (4.75, 4.8), meaning both x and y could be in this overlapping region. This intersection is guaranteed to exist. Choices B and C incorrectly assume one is always greater based on interval endpoints, while choice D fails to recognize the definite overlap.
Which of the following values is the greatest?
Answer: -1.001 When comparing negative numbers, the number with the smallest absolute value (the one closest to zero) is the greatest. On a number line, -1.001 is to the right of the other numbers, making it the greatest value.
On a number line, point X is at -2.4 and point Y is at 1.8. What is the distance between X and Y?
Answer: 4.2 The distance between two points on a number line is the absolute value of their difference. Alternatively, since one point is negative and one is positive, add their absolute values. The distance from -2.4 to 0 is 2.4, and the distance from 0 to 1.8 is 1.8. The total distance is $2.4 + 1.8 = 4.2$ .
A baker needs 3.75 pounds of sugar. The scale measures in increments of tenths of a pound. Which of the following shows a weight that is less than required, but is the closest possible reading on the scale?
Answer: 3.7 pounds The required amount is 3.75 pounds. The scale measures in tenths (3.5, 3.6, 3.7, 3.8, etc.). We need a reading that is less than 3.75. Both 3.6 and 3.7 are less. To find which is closest, we compare the differences: $3.75 - 3.7 = 0.05$ and $3.75 - 3.6 = 0.15$ . The smaller difference is with 3.7, so it is the closest reading that is less than the required amount.
The number $3.\overline{45}$ (where $45$ repeats) is positioned on a number line. Using place value reasoning, which of the following is the most accurate comparison with $3\frac{5}{11}$ ?
Answer: $3.\overline{45} = 3\frac{5}{11}$ because both equal exactly $\frac{38}{11}$ Convert $3.\overline{45}$ to a fraction: Let $x = 3.\overline{45} = 3.454545...$ . Then $100x = 345.454545...$ and $100x - x = 345.454545... - 3.454545... = 342$ , so $99x = 342$ and $x = \frac{342}{99} = \frac{38}{11}$ . Also, $3\frac{5}{11} = \frac{33 + 5}{11} = \frac{38}{11}$ . Therefore they are equal. Choices A and D make computational errors, while choice B incorrectly suggests precision affects value.
A point is located at $\frac{8}{3}$ on a number line. Which of the following numbers is closest to this point?
Answer: 2.70 First, convert $\frac{8}{3}$ to a decimal. $8 \div 3 = 2.666...$ . Now find the distance between this value and each option. The difference between 2.666... and 2.60 is 0.066.... The difference between 2.666... and 2.70 is 0.033.... The difference between 2.666... and 2.75 is 0.083.... The difference between 2.666... and 2.80 is 0.133.... The smallest difference is with 2.70.
Which of the following expressions is equivalent to the number 305.08?
Answer: $3 \times 100 + 5 \times 1 + 8 \times \frac{1}{100}$ The number 305.08 can be broken down by place value: the 3 is in the hundreds place ( $3 \times 100$ ), the 0 is in the tens place, the 5 is in the ones place ( $5 \times 1$ ), the 0 is in the tenths place, and the 8 is in the hundredths place ( $8 \times \frac{1}{100}$ ). Summing these parts gives the correct expression.
Three packages have weights of 2.3 kilograms, 2.25 kilograms, and 2.035 kilograms. Which of the following correctly lists the weights in order from heaviest to lightest?
Answer: 2.3, 2.25, 2.035 To compare the decimals, it can be helpful to add trailing zeros so they have the same number of decimal places: 2.300, 2.250, 2.035. Comparing these values, 2.300 is the largest (heaviest), followed by 2.250, and then 2.035 is the smallest (lightest).
In the number 62.174, the value of the digit 7 is subtracted from the value of the digit 2. What is the result?
Answer: 1.93 The value of the digit 2 is 2 (since it's in the ones place). The value of the digit 7 is 0.07 (since it's in the hundredths place). The subtraction is $2 - 0.07 = 1.93$ .
How many times greater is the value represented by the digit 5 in the number 52.4 than the value represented by the digit 4 in the same number?
Answer: 125 In 52.4, the value of the digit 5 is 50 (since it is in the tens place). The value of the digit 4 is 0.4 (since it is in the tenths place). To find how many times greater 50 is than 0.4, we divide: $50 \div 0.4 = 500 \div 4 = 125$ .
On a number line, point M is at 4.2. Point N is to the right of M, and the distance between M and N is 0.5. Point P is located halfway between M and N. What is the value at point P?
Answer: 4.45 First, find the location of point N. It is to the right of M (4.2) by a distance of 0.5, so $N = 4.2 + 0.5 = 4.7$ . Point P is halfway between M (4.2) and N (4.7). The distance between them is 0.5. Half of this distance is $0.5 \div 2 = 0.25$ . Adding this to M gives the location of P: $4.2 + 0.25 = 4.45$ .
If the number 72.84 is divided by 100, the digit 8 will be in which place value position in the new number?
Answer: Thousandths Dividing 72.84 by 100 shifts the decimal point two places to the left, resulting in the number 0.7284. In this new number, the digit 7 is in the tenths place, 2 is in the hundredths place, and 8 is in the thousandths place.
The numbers on a number line increase by the same amount between each consecutive tick mark. If the first two tick marks are at 4.8 and 5.15, what is the value of the fourth tick mark?
Answer: 5.85 First, find the amount of increase between tick marks: 5.15 - 4.8 = 0.35. Starting from the first mark at 4.8: second mark is at 5.15, third mark is at 5.15 + 0.35 = 5.50, and fourth mark is at 5.50 + 0.35 = 5.85.
On a number line, point P is located at 1.6 and point Q is located at 2.1. Point R is located exactly halfway between P and Q. What is the location of point R?
Answer: 1.85 First, find the distance between P and Q: $2.1 - 1.6 = 0.5$ . The halfway point is half of this distance, which is $0.5 \div 2 = 0.25$ . Add this distance to the starting point P to find R: $1.6 + 0.25 = 1.85$ .
The number $2\frac{3}{8}$ is located on a number line. Which of the following numbers requires moving the farthest distance from $2\frac{3}{8}$ along the number line?
Answer: $2\frac{1}{3}$ because the distance is approximately $0.042$ units Convert $2\frac{3}{8} = 2.375$ . Calculate distances: |2.4 - 2.375| = 0.025, |2.35 - 2.375| = 0.025, |2.4 - 2.375| = 0.025. For $2\frac{1}{3} = 2.333...$ : |2.333... - 2.375| = 0.041... ≈ 0.042. This is the largest distance. The other choices all have the same distance of 0.025 units.

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