Comparing & Ordering Decimals - Accuplacer Arithmetic

1. Which of the following values is closest to 0.6?
   Answer: 0.601 To find the value closest to 0.6, calculate the absolute difference between 0.6 and each option: |0.6 - 0.59| = 0.01; |0.6 - 0.601| = 0.001; |0.6 - 0.598| = 0.002; |0.6 - 0.61| = 0.01. The smallest difference is 0.001, which corresponds to the value 0.601.
2. A pharmacist needs to measure a liquid medicine dose that is greater than 1.25 mL but less than 1.3 mL. Which of the following doses is acceptable?
   Answer: 1.28 mL The question asks for a number between 1.25 and 1.3. A) 1.248 is less than 1.25. B) 1.25 is not greater than 1.25. C) 1.28 is greater than 1.25 and less than 1.3. D) 1.3 is not less than 1.3. Therefore, 1.28 mL is the only acceptable dose.
3. A jeweler measures gem weights: $2.034$ carats, $2.304$ carats, $2.043$ carats, and $2.340$ carats. If arranged from heaviest to lightest, which gem weight appears in the second position?
   Answer: $2.304$ carats Ordering from heaviest to lightest: 2.340 > 2.304 > 2.043 > 2.034. The second position contains 2.304 carats. Choice A (2.034) is in fourth position. Choice C (2.043) is in third position. Choice D (2.340) is in first position.
4. A bank offers four different savings accounts with the following annual interest rates: 0.02, 0.015, 0.021, and 0.019. To earn the most interest, which rate should a customer choose?
   Answer: 0.021 To earn the most interest, the customer should choose the account with the highest interest rate. Comparing the decimal values: 0.015, 0.019, 0.020 (same as 0.02), and 0.021. The largest of these numbers is 0.021.
5. A mechanic is checking the gap of a spark plug. The acceptable range is from 0.7 millimeters to 0.8 millimeters, inclusive. Which of the following measurements is too small?
   Answer: 0.695 mm The acceptable range is [0.7, 0.8]. A measurement is 'too small' if it is less than 0.7. A) 0.701 is in the range. B) 0.79 is in the range. C) 0.8 is the maximum acceptable value, not too small. D) 0.695 is less than 0.7, so it is too small.
6. A pharmacist needs to dispense medication doses in ascending order. The prescribed doses are $2.75$ mg, $2.705$ mg, $2.7$ mg, and $2.715$ mg. After arranging them correctly, which dose comes third in the sequence?
   Answer: $2.715$ mg Converting to thousandths: 2.700, 2.705, 2.715, 2.750. In ascending order: 2.7, 2.705, 2.715, 2.75. The third position is 2.715 mg. Choice A (2.7) is first. Choice B (2.705) is second. Choice D (2.75) is fourth in the sequence.
7. A machinist measures bolt diameters: $0.249$ inches, $0.25$ inches, $0.2495$ inches, and $0.2505$ inches. Bolts with diameters between $0.248$ and $0.251$ inches (inclusive) pass inspection. Which statement about the inspection results is correct?
   Answer: All four bolts pass inspection Checking each diameter against 0.248 ≤ diameter ≤ 0.251: 0.249 (passes), 0.25 (passes), 0.2495 (passes), 0.2505 (passes). All measurements fall within the acceptable range. Choice A undercounts by one. Choices C and D significantly undercount the passing bolts.
8. A quality control inspector measures the thickness of metal sheets. The measurements are: $0.0625$ inches, $0.063$ inches, $0.0619$ inches, and $0.06205$ inches. If the sheets must be arranged from thinnest to thickest for packaging, what is the correct order?
   Answer: $0.0619$, $0.06205$, $0.0625$, $0.063$ To compare decimals, align decimal places: 0.0619 < 0.06205 < 0.0625 < 0.063. Choice A correctly orders from thinnest to thickest. Choice B incorrectly places 0.0625 before 0.06205. Choice C incorrectly places 0.06205 before 0.0619. Choice D orders from thickest to thinnest instead of thinnest to thickest.
9. On a number line, which of the following values is located to the right of -4.6 but to the left of -4.5?
   Answer: -4.55 On a number line, 'to the right' means greater than, and 'to the left' means less than. The question is asking for a number x such that -4.6 < x < -4.5. A) -4.65 is less than -4.6. B) -4.55 is greater than -4.6 and less than -4.5. C) -4.7 is less than -4.6. D) -4.4 is greater than -4.5. The correct value is -4.55.
10. Three runners complete a race with times of $12.045$ seconds, $12.4$ seconds, and $12.05$ seconds. A fourth runner's time is greater than the fastest time but less than $12.046$ seconds. Which decimal could represent the fourth runner's time?
    Answer: $12.0451$ The fastest time is 12.045 seconds. The fourth runner's time must be between 12.045 and 12.046 (exclusive). Only 12.0451 satisfies 12.045 < 12.0451 < 12.046. Choice B (12.044) is less than 12.045. Choice C (12.0459) is greater than 12.046. Choice D (12.046) equals the upper bound, not less than it.
11. Which of the following statements is true?
    Answer: 0.125 = 1/8 To evaluate each statement, convert the fractions to decimals. A) 2/5 = 0.4. The statement 0.45 < 0.4 is false. B) 3/4 = 0.75. The statement 0.7 > 0.75 is false. C) 1/8 = 0.125. The statement 0.125 = 0.125 is true. D) 2/3 = 0.666... The statement 0.6 > 0.666... is false.
12. A delivery service sorts packages by weight. Package weights are $3.08$ kg, $3.008$ kg, $3.8$ kg, and $3.080$ kg. How many packages weigh more than $3.07$ kg but less than $3.1$ kg?
    Answer: 2 We need weights where 3.07 < weight < 3.1. Checking each: 3.08 kg (yes, 3.07 < 3.08 < 3.1), 3.008 kg (no, 3.008 < 3.07), 3.8 kg (no, 3.8 > 3.1), 3.080 kg (yes, same as 3.08, so 3.07 < 3.08 < 3.1). Two packages meet the criteria. Other choices miscount the packages in the specified range.
13. A biologist measured the lengths of four insects in centimeters: 0.508 cm, 0.58 cm, 0.499 cm, and 0.5 cm. Which of these is the smallest measurement?
    Answer: 0.499 cm To compare the decimals, align the decimal points and add trailing zeros so they have the same number of digits: 0.508, 0.580, 0.499, and 0.500. Comparing the numbers, the smallest value is 0.499.
14. A lab technician measures liquid volumes: $15.6$ mL, $15.06$ mL, $15.006$ mL, and $15.66$ mL. Which two volumes have the greatest difference between them?
    Answer: $15.66$ mL and $15.006$ mL When comparing decimal numbers, you need to carefully examine each digit's place value to determine which numbers are furthest apart. The key is to align the decimal places and subtract to find the actual differences. Let's find the difference for each pair. First, organize the numbers by size: $15.006$ (smallest), $15.06$, $15.6$, and $15.66$ (largest). The greatest difference will be between the largest and smallest values: $15.66 - 15.006 = 0.654$ mL. Let's verify by checking the other options. Option A gives us $15.6 - 15.006 = 0.594$ mL. Option B yields $15.6 - 15.06 = 0.54$ mL. Option C produces $15.66 - 15.06 = 0.6$ mL. Comparing all differences, $0.654$ mL is indeed the largest, confirming that option D is correct. Option A is wrong because while $15.6$ and $15.006$ are far apart, $15.66$ is even larger than $15.6$, creating a greater gap. Option B is incorrect because both numbers are relatively close to each other in the middle range. Option C fails because although $15.66$ is the largest number, $15.06$ isn't the smallest—$15.006$ is smaller. When finding the greatest difference between numbers in a set, always identify the maximum and minimum values first. This saves time and prevents calculation errors. Also, be extra careful with decimal place values—$15.06$ and $15.006$ look similar but differ significantly in the hundredths place.
15. Which of the following lists of numbers is ordered from least to greatest?
    Answer: -1.5, -1.45, -1.05, -1.04 When ordering negative numbers, the number with the largest absolute value is the least. Comparing the numbers: -1.5 is the smallest. Then comes -1.45. Next is -1.05. The greatest number is -1.04. The correct order from least to greatest is -1.5, -1.45, -1.05, -1.04.
16. Three of the following numbers are between 3.1 and 3.2. Which number is NOT in this range?
    Answer: 3.099 A number between 3.1 and 3.2 must be greater than 3.1 and less than 3.2. A) 3.105 is in the range. B) 3.15 is in the range. C) 3.099 is less than 3.1, so it is not in the range. D) 3.199 is in the range. The number not in the range is 3.099.
17. What is the result when the numbers -2.1, 1.2, -1.2, and 2.01 are arranged in order from least to greatest?
    Answer: -2.1, -1.2, 1.2, 2.01 To order from least to greatest, start with the most negative number. -2.1 is less than -1.2. Then order the positive numbers. 1.2 is less than 2.01. The complete order is -2.1, -1.2, 1.2, 2.01.
18. Consider two calculations: Calculation P: 2.5 - 0.05 Calculation Q: 2.4 + 0.055 Which of the following statements correctly compares the results?
    Answer: The result of P is less than the result of Q. First, perform each calculation. Calculation P: 2.5 - 0.05 = 2.45. Calculation Q: 2.4 + 0.055 = 2.455. Next, compare the two results: 2.45 and 2.455. Since 2.450 is less than 2.455, the result of P is less than the result of Q.
19. A list of numbers is shown: 5.05, 5.5, 5.055, 5.505. If these numbers are arranged in descending order (greatest to least), what is the third number in the new list?
    Answer: 5.055 To arrange the numbers in descending order, find the largest value first. It's helpful to add trailing zeros to compare: 5.050, 5.500, 5.055, 5.505. The descending order is: 1st: 5.505, 2nd: 5.500 (or 5.5), 3rd: 5.055, 4th: 5.050 (or 5.05). The third number in this ordered list is 5.055.
20. A scientist records temperatures: $-1.205°C$, $-1.25°C$, $-1.2°C$, and $-1.21°C$. Which temperature is closest to zero?
    Answer: $-1.2°C$ When comparing negative numbers, remember that the closer a negative number is to zero, the larger its value. Think of a number line: $-1$ is closer to zero than $-2$, so $-1 > -2$. To find which temperature is closest to zero, you need to determine which has the smallest absolute value (distance from zero). Let's compare the absolute values: $|-1.205| = 1.205$, $|-1.25| = 1.25$, $|-1.2| = 1.2$, and $|-1.21| = 1.21$. The correct answer is C) $-1.2°C$ because $1.2$ is the smallest absolute value, making $-1.2$ closest to zero. Here's why the other answers are incorrect: A) $-1.21°C$ has an absolute value of $1.21$, which is larger than $1.2$, so it's further from zero than option C. B) $-1.205°C$ has an absolute value of $1.205$, also larger than $1.2$. D) $-1.25°C$ has the largest absolute value at $1.25$, making it the furthest from zero. A common mistake is thinking that more decimal places means a smaller number, or confusing which negative number is "larger." Remember: when comparing negative decimals, the one with the smallest absolute value is closest to zero and therefore the "largest" negative number. Always convert to absolute values when finding distance from zero, or visualize the numbers on a number line.

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