Decimals & Fractions: Level 2

Solution (D): 1. Find \(x\): Sum first. \(2.95 + 3.95 + 1.05 = 7.95\). Round to the tenths place: \(x = 8.0\). 2. Find \(y\): Round first. \(a = 2.95 \to 3.0\) \(b = 3.95 \to 4.0\) \(c = 1.05 \to 1.1\) Sum the rounded values: \(y = 3.0 + 4.0 + 1.1 = 8.1\). 3. Find \(y – x\): \(8.1 – 8.0 = 0.1\).

Solution (B): The repeating pattern is “123456”, which has a length of 6 digits. We divide the position we want (50) by the length of the pattern (6). \(50 \div 6 = 8\) with a remainder of 2. A remainder of 2 means it is the 2nd digit in the pattern, which is 2.

Solution (C): To round to 4.5, the original number must be in the range [4.45, 4.55). (A) 4.4 is too small. (B) 4.44 is too small. (C) 4.49 is in the range. (D) 4.55 is not in the range (it rounds to 4.6). (E) 4.56 is too large.

Solution (C): Convert all values to decimals to compare. A) \( 5 \div 6 = 0.8333… \) B) \( 0.8 = 0.8000 \) C) \( 7 \div 8 = 0.8750 \) D) \( 0.83 = 0.8300 \) E) \( 11 \div 13 \approx 0.846… \) Comparing the decimals, 0.8750 is the largest.

Solution (A): We need to find “one-half of one-third”. “Of” means multiply. \( \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \).

Solution (D): 1. Solve the parenthesis: \( \frac{1}{2} – \frac{1}{3} \). Find a common denominator (6). \( \frac{3}{6} – \frac{2}{6} = \frac{1}{6} \). 2. Perform the division: \( \frac{1}{6} \div \frac{1}{4} \). Invert and multiply. \( \frac{1}{6} \times \frac{4}{1} = \frac{4}{6} \). 3. Simplify the result: \( \frac{4}{6} = \frac{2}{3} \).

Solution (B): The digits “12” do not repeat. The pattern “34” (length 2) begins at the 3rd decimal place. We want the 80th digit. The first 2 are fixed. We need to find the \( (80 – 2) = 78\text{th} \) digit of the “34” pattern. Divide 78 by the pattern length (2): \( 78 \div 2 = 39 \) with a remainder of 0. A remainder of 0 means it is the last (2nd) digit of the pattern, which is 4.

Solution (D): A fraction will terminate if and only if its denominator (in simplest form) contains only 2s and/or 5s as prime factors. (A) \( 12 = 2^2 \times 3 \). (Repeats) (B) \( 9 = 3^2 \). (Repeats) (C) \( 7 = 7 \). (Repeats) (D) \( 8 = 2^3 \). (Terminates) (E) \( 6 = 2 \times 3 \). (Repeats)

Solution (A): The range of numbers that round to 5 is [4.5, 5.5). The given range for \(x\) is (4.7, 5.3), which is \(\{ 4.7 < x < 5.3 \}\). Since this entire range is contained *within* the [4.5, 5.5) range, any number \(x\) in this range *must* round to 5.

Solution (B): We test each option for \(y – x = 1\). (A) \(a=2.4, b=3.4\). \(x\): Sum = 5.8 \(\to\) 6. \(y\): \(2 + 3 = 5\). \(y-x = 5-6 = -1\). (B) \(a=2.5, b=3.5\). \(x\): Sum = 6.0 \(\to\) 6. \(y\): \(3 + 4 = 7\). \(y-x = 7-6 = 1\). This is the answer. (C) \(a=2.1, b=3.1\). \(x\): Sum = 5.2 \(\to\) 5. \(y\): \(2 + 3 = 5\). \(y-x = 5-5 = 0\).