Praxis Core Number and Quantity Summary

1. The common denominator of 4 and 3 is 12.
2. 3/4 = 9/12 and 2/3 = 8/12.
3. 9/12 + 8/12 = 17/12 (which is 1 and 5/12).

Check: 0.75 + 0.667 ≈ 1.417, and 17 ÷ 12 ≈ 1.417.

Wrong choices

• "5/7": This adds the tops and the bottoms (3 + 2 and 4 + 3). Fractions need a common denominator first.
• "11/12": This changes 3/4 to 9/12 correctly but writes 2/3 as 2/12 without multiplying the top by 4.
• "1/2": This is the product 3/4 × 2/3 = 6/12, not the sum.
• "1/12": This is the difference 9/12 − 8/12, not the sum.

1. 35% = 0.35.
2. 0.35 × 240 = 84.

Check: 10% of 240 is 24, so 30% is 72 and 5% is 12; 72 + 12 = 84.

Wrong choices

• "8.4": This moves the decimal one place too far. 35% is 0.35, not 0.035.
• "156": This is 240 − 84, the part that is NOT 35%.
• "840": This uses 3.5 instead of 0.35.
• "275": This simply adds 35 and 240.

1. Total parts: 3 + 5 = 8.
2. Each part: 40 ÷ 8 = 5 students.
3. Girls: 5 parts × 5 = 25.

Check: boys = 3 × 5 = 15, and 15 + 25 = 40.

Wrong choices

• "15": 15 is the number of boys, not girls.
• "24": This finds 3/5 of 40. The girls are 5 out of 8 total parts, not 3/5.
• "8": 8 is the total number of ratio parts (3 + 5), not the number of girls.
• "20": This splits the class in half, ignoring the 3-to-5 ratio.

1. Price per pound = $9.90 ÷ 6.
2. 9.90 ÷ 6 = 1.65.

Check: 6 × $1.65 = $9.90.

Wrong choices

• "$1.50": This divides only the whole dollars ($9 ÷ 6) and ignores the 90 cents.
• "$0.61": This divides 6 by 9.90 (pounds per dollar), which is backward.
• "$3.90": This subtracts 6 from 9.90 instead of dividing.
• "$59.40": This multiplies 9.90 by 6 instead of dividing.

1. Increase: $75 − $60 = $15.
2. Divide by the original: 15 ÷ 60 = 0.25.
3. 0.25 = 25%.

Check: 25% of $60 is $15, and $60 + $15 = $75.

Wrong choices

• "20%": This divides the $15 increase by the NEW price ($75). Percent change uses the original price.
• "15%": 15 is the dollar increase, not the percent increase.
• "80%": This is 60 ÷ 75, which compares the two prices but is not the change.
• "125%": This is the new price as a percent of the old price (75 ÷ 60), not the increase.

1. Discount: 25% of $80 = $20, so the sale price is $60.
2. Tax: 8% of $60 = $4.80.
3. Total: $60 + $4.80 = $64.80.

Check: $80 × 0.75 × 1.08 = $64.80.

Wrong choices

• "$66.40": This takes 25% − 8% = 17% off. Discount and tax are applied one after the other, not combined.
• "$60.00": This is the sale price before tax.
• "$53.60": This takes off 25% + 8% = 33%. The tax adds to the price; it does not reduce it.
• "$86.40": This adds the tax to the original price and forgets the 25% discount.

1. Set up: 3/24 = x/40.
2. Cross-multiply: 24x = 3 × 40 = 120.
3. x = 120 ÷ 24 = 5 cups.

Check: 1 cup makes 8 cookies, and 5 × 8 = 40.

Wrong choices

• "1.8 cups": This sets up the proportion upside down (24 × 3 ÷ 40).
• "8 cups": 8 is the number of cookies per cup (24 ÷ 3), not the cups needed.
• "13.3 cups": This divides 40 by 3, mixing up cookies and cups.
• "19 cups": This adds 16 (the extra cookies) to 3 cups. A proportion multiplies; it does not add.

1. Division and multiplication first: 9 ÷ 3 = 3 and 2 × 5 = 10.
2. Then left to right: 15 − 3 + 10 = 12 + 10 = 22.

Check: 15 − 3 = 12, and 12 + 10 = 22.

Wrong choices

• "20": This works strictly left to right: 15 − 9 = 6, 6 ÷ 3 = 2, 2 + 2 = 4, 4 × 5 = 20. Division and multiplication must come first.
• "12": This does 15 − 9 first and then divides: (15 − 9) ÷ 3 + 10 = 12.
• "2": This adds 3 + 10 before subtracting: 15 − 13 = 2. Addition and subtraction are done left to right.
• "6": This adds 3 + 2 first, as if there were parentheses around it.

1. 3.5 × 12 = 42.
2. So the board is 42 inches long.

Check: 3 feet = 36 inches and half a foot = 6 inches; 36 + 6 = 42.

Wrong choices

• "41 inches": This treats 3.5 feet as 3 feet 5 inches. Half a foot is 6 inches, not 5.
• "36 inches": This converts only the 3 whole feet and drops the half foot.
• "35 inches": This just moves the decimal point; feet and inches are not related by 10.
• "0.29 inches": This divides 3.5 by 12. To go from a larger unit to a smaller unit, multiply.

1. −7 + 12 = 5.
2. 5 − (−5) = 5 + 5 = 10.

Check: on a number line, start at −7, move right 12 to 5, then right 5 more to 10.

Wrong choices

• "0": This treats − (−5) as −5. Subtracting a negative number is the same as adding it.
• "24": This ignores all the negative signs and adds 7 + 12 + 5.
• "−10": This flips every sign: 7 − 12 − 5 = −10.
• "−14": This subtracts 12 instead of adding it: −7 − 12 + 5 = −14.