GED Math Number Sense and Operations Summary

1. Parentheses first: 2 + 4 = 6.
2. Multiply and divide from left to right: 3 × 6 = 18, then 18 ÷ 9 = 2.
3. Subtract last: 18 − 2 = 16.

Check: On a calculator that follows the order of operations, 18 − 3 × 6 ÷ 9 also gives 16.

Wrong choices

• 10: 10 works strictly left to right: (18 − 3) × 6 ÷ 9. Subtraction must wait until multiplication and division are done.
• 0: 0 subtracts 3 × 6 from 18 first and then divides by 9; the division belongs to the 3 × 6 part only.
• 2: 2 is only the value of 3 × 6 ÷ 9; it still has to be subtracted from 18.

1. Change the fractions to decimals: 3/5 = 0.6 and 5/8 = 0.625.
2. Now compare four decimals: 0.65, 0.625, 0.58, 0.6.
3. From least to greatest: 0.58, 0.6, 0.625, 0.65.
4. Write them in their original form: 0.58, 3/5, 5/8, 0.65.

Check: 0.58 < 0.60 < 0.625 < 0.65, so each number is bigger than the one before it.

Wrong choices

• 3/5, 5/8, 0.58, 0.65: 3/5 (0.6) is larger than 0.58, so 3/5 cannot come first.
• 0.58, 0.65, 3/5, 5/8: This lists the decimals first and the fractions after, but 3/5 (0.6) and 5/8 (0.625) are smaller than 0.65.
• 0.65, 5/8, 3/5, 0.58: This is the right order but backward: it goes from greatest to least.

1. Write 3/4 with a denominator of 8: 3/4 = 6/8, so 8 3/4 = 8 6/8.
2. Subtract the whole numbers: 8 − 2 = 6.
3. Subtract the fractions: 6/8 − 5/8 = 1/8.
4. The board left is 6 1/8 feet.

Check: 6 1/8 + 2 5/8 = 8 6/8 = 8 3/4 feet, the original length.

Wrong choices

• 11 3/8 feet: 11 3/8 feet comes from adding the two lengths instead of subtracting.
• 6 2/4 feet: 6 2/4 feet comes from subtracting the tops and the bottoms of the fractions separately, which is not how fractions work.
• 6 3/8 feet: 6 3/8 feet treats 3/4 as a whole 1 before subtracting 5/8; 3/4 must be changed to 6/8, not 1.

1. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
2. Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
3. The largest number in both lists is 12.

Check: 36 ÷ 12 = 3 and 60 ÷ 12 = 5, both whole numbers, and 3 and 5 share no other factor.

Wrong choices

• 6: 6 divides both numbers, but it is not the largest; 12 also divides both.
• 180: 180 is the least common multiple, not the greatest common factor.
• 4: 4 divides both numbers, but it is not the largest common factor.

1. Rise = later temperature − earlier temperature.
2. 15 − (−8) = 15 + 8 = 23.
3. The temperature rose 23 degrees.

Check: Count on a number line: from −8 up to 0 is 8 degrees, and from 0 up to 15 is 15 more. 8 + 15 = 23.

Wrong choices

• 7 degrees: 7 degrees comes from 15 − 8, which ignores that −8 is below zero.
• −23 degrees: A rise is a positive change; −23 has the wrong sign.
• 8 degrees: 8 degrees only counts the rise from −8 up to 0.

1. Regular pay: 40 × $18.50 = $740.00.
2. Overtime rate: 1.5 × $18.50 = $27.75 per hour.
3. Overtime hours: 46 − 40 = 6. Overtime pay: 6 × $27.75 = $166.50.
4. Total: $740.00 + $166.50 = $906.50.

Check: Think of it as 46 regular hours ($851.00) plus an extra half rate for 6 hours (6 × $9.25 = $55.50). $851.00 + $55.50 = $906.50.

Wrong choices

• $851.00: $851.00 pays all 46 hours at the regular rate and leaves out the overtime extra.
• $860.00: $860.00 adds $1.50 to the hourly rate instead of multiplying the rate by 1.5.
• $1,276.50: $1,276.50 pays all 46 hours at the overtime rate; only the 6 hours over 40 get the higher rate.

1. List multiples of 12: 12, 24, 36, 48 ...
2. List multiples of 18: 18, 36, 54 ...
3. The least common multiple is 36 minutes.
4. 8:00 a.m. + 36 minutes = 8:36 a.m.

Check: 36 ÷ 12 = 3 trips and 36 ÷ 18 = 2 trips, so both buses are at the terminal at 8:36.

Wrong choices

• 8:30 a.m.: 8:30 a.m. adds 12 and 18; 30 minutes is not a multiple of 12 or 18.
• 8:06 a.m.: 8:06 a.m. uses the greatest common factor (6), but neither bus leaves after only 6 minutes.
• 11:36 a.m.: 11:36 a.m. multiplies 12 × 18 = 216 minutes; that is a common time, but not the first one.

1. Write 30 as 3 × 10¹.
2. Multiply the front numbers: 2.4 × 3 = 7.2.
3. Multiply the powers of 10: 10⁵ × 10¹ = 10⁶.
4. The total is 7.2 × 10⁶ bolts.

Check: 240,000 × 30 = 7,200,000, which is 7.2 × 10⁶.

Wrong choices

• 7.2 × 10⁵: 7.2 × 10⁵ multiplies by 3 instead of 30; it forgets the factor of 10 in 30.
• 7.2 × 10⁷: 7.2 × 10⁷ adds two to the power of 10, but 30 has only one factor of 10.
• 2.4003 × 10⁵: 2.4003 × 10⁵ adds 30 to 240,000 instead of multiplying.

1. Power of a power: (2³)² = 2^(3 × 2) = 2⁶.
2. Same base multiplied: 2⁶ × 2⁻⁴ = 2^(6 + (−4)) = 2².
3. 2² = 4.

Check: (2³)² = 8² = 64, and 2⁻⁴ = 1/16. 64 × 1/16 = 4.

Wrong choices

• 2: 2 adds the exponents 3 and 2 inside the power (getting 2⁵) instead of multiplying them.
• 32: 32 squares the exponent 3 to get 2⁹; the rule is to multiply 3 × 2.
• 1,024: 1,024 treats −4 as +4 and gets 2¹⁰.

1. Side of the square rug: √81 = 9 feet.
2. Edge of the cube box: ∛64 = 4 feet (because 4 × 4 × 4 = 64).
3. Difference: 9 − 4 = 5 feet.

Check: 9 × 9 = 81 square feet and 4 × 4 × 4 = 64 cubic feet.

Wrong choices

• 1 foot: 1 foot uses the square root of 64 (8) for the box; a cube needs the cube root.
• 17 feet: 17 feet subtracts the area and volume (81 − 64) instead of comparing lengths.
• 13 feet: 13 feet adds the two lengths instead of finding the difference.