Perform operations with whole numbers, fractions, and decimals - TACHS Math

1. Compute $2-\dfrac{3}{8}\times4.$
   Answer: $\dfrac{1}{2}$ When you see an expression with both multiplication and subtraction, remember that order of operations (PEMDAS) requires you to perform multiplication before subtraction. Let's work through this step by step. First, calculate $\dfrac{3}{8} \times 4$. When multiplying a fraction by a whole number, multiply the numerator by the whole number: $\dfrac{3}{8} \times 4 = \dfrac{3 \times 4}{8} = \dfrac{12}{8}$. This simplifies to $\dfrac{3}{2}$ or $1\dfrac{1}{2}$. Now subtract this result from 2: $2 - \dfrac{3}{2}$. Convert 2 to a fraction with denominator 2: $\dfrac{4}{2} - \dfrac{3}{2} = \dfrac{1}{2}$. This confirms answer choice D. Looking at the wrong answers: Choice A ($1$) likely comes from incorrectly calculating $2 - \dfrac{3}{8} = \dfrac{16}{8} - \dfrac{3}{8} = \dfrac{13}{8}$, then somehow getting confused in the arithmetic. Choice B ($\dfrac{5}{8}$) results from the error $2 - \dfrac{3}{8} = \dfrac{16}{8} - \dfrac{3}{8} = \dfrac{13}{8}$, but then incorrectly trying to account for the multiplication by making an arithmetic mistake. Choice C ($\dfrac{3}{4}$) comes from various computational errors, possibly mixing up the order of operations. The key strategy here is to always follow PEMDAS strictly. When you see mixed operations, identify what needs to be done first, then work systematically through each step. Double-check your fraction arithmetic, especially when converting between mixed numbers and improper fractions.
2. Using the order of operations, compute $6.4-2.1\times3.$
   Answer: 0.1 When you encounter an expression with multiple operations, you must follow the order of operations (PEMDAS/BODMAS). This means you perform multiplication and division before addition and subtraction, working from left to right. Let's work through $6.4-2.1\times3$ step by step. First, identify the operations: subtraction and multiplication. Since multiplication comes before subtraction in the order of operations, you must calculate $2.1\times3$ first: $2.1\times3=6.3$. Now substitute this result back into the original expression: $6.4-6.3=0.1$. Therefore, the answer is D) 0.1. The wrong answers reveal common mistakes. Choice A) 8.5 likely comes from adding instead of subtracting: $6.4+2.1=8.5$, ignoring the multiplication entirely. Choice B) 12.9 results from incorrectly working left to right without following order of operations: $(6.4-2.1)\times3=4.3\times3=12.9$. This treats subtraction as having equal priority to multiplication and groups the subtraction first. Choice C) 4.3 represents the intermediate step $6.4-2.1=4.3$, but stops before completing the multiplication. Remember that order of operations questions often include answer choices that represent common procedural errors. Always identify all operations first, then systematically apply PEMDAS. When you see multiplication or division mixed with addition or subtraction, resist the urge to work strictly left to right—complete the multiplication and division operations first, then handle addition and subtraction from left to right.
3. Add the decimals: $45.067+3.9+0.08.$
   Answer: 49.047 When adding decimals, the key is proper alignment of place values. You need to line up the decimal points vertically so that tenths align with tenths, hundredths with hundredths, and so on. Let's set up $45.067 + 3.9 + 0.08$ with proper alignment: 45.067 3.900 0.080 ------ 49.047 Notice how I added zeros to make the place values clear: $3.9$ becomes $3.900$ and $0.08$ becomes $0.080$. Now we can add column by column from right to left: $7 + 0 + 0 = 7$ (thousandths), $6 + 0 + 8 = 14$ (hundredths, write 4 and carry 1), $0 + 9 + 0 + 1 = 10$ (tenths, write 0 and carry 1), $5 + 3 + 0 + 1 = 9$ (ones), and $4 + 0 + 0 = 4$ (tens). Looking at the wrong answers: Choice A (48.147) likely comes from misaligning the decimal points or incorrectly adding the tenths place. Choice B (48.957) suggests an error in both the ones place (getting 8 instead of 9) and the hundredths place. Choice C (49.837) indicates confusion in the decimal portion, possibly from misplacing digits during the addition process. The correct answer is D (49.047). Study tip: Always write out decimal addition vertically with aligned decimal points, and don't hesitate to add trailing zeros to make place values crystal clear. This prevents the most common decimal addition errors.
4. Simplify $\dfrac{7}{9}\times\dfrac{6}{14}$ to lowest terms.
   Answer: $\dfrac{1}{3}$ When multiplying fractions, you multiply the numerators together and the denominators together, then simplify to lowest terms. However, you can make your work easier by canceling common factors before multiplying. Let's set up the multiplication: $\frac{7}{9} \times \frac{6}{14}$ Before multiplying, look for common factors you can cancel. Notice that 7 and 14 share a factor of 7, and 6 and 9 share a factor of 3: $\frac{7}{9} \times \frac{6}{14} = \frac{\cancel{7}^1}{9} \times \frac{6}{\cancel{14}^2} = \frac{1 \times 6}{9 \times 2}$ You can also cancel the 6 in the numerator with the 9 in the denominator (they share a factor of 3): $\frac{1 \times \cancel{6}^2}{{\cancel{9}^3} \times 2} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6} = \frac{1}{3}$ This confirms answer choice C is correct. Let's examine why the other answers are wrong. Answer A ($\frac{2}{3}$) would result if you incorrectly canceled factors or made arithmetic errors in your cancellation. Answer B ($\frac{1}{2}$) might occur if you confused which numbers to cancel or switched numerators and denominators during simplification. Answer D ($\frac{5}{7}$) doesn't relate to any logical step in this problem and likely represents a distractor. Study tip: Always look for common factors to cancel before multiplying fractions—it prevents you from working with large numbers and reduces the chance of arithmetic errors. Cross-canceling (canceling factors diagonally) is especially helpful and saves time.
5. Find the product $1\dfrac{2}{3}\times2\dfrac{1}{2}$ and express it as a mixed number in simplest form.
   Answer: $4\dfrac{1}{6}$ When you're multiplying mixed numbers, you need to convert them to improper fractions first, then multiply and convert back to a mixed number. Start by converting each mixed number to an improper fraction. For $1\frac{2}{3}$: multiply the whole number by the denominator and add the numerator: $1 \times 3 + 2 = 5$, so you get $\frac{5}{3}$. For $2\frac{1}{2}$: $2 \times 2 + 1 = 5$, giving you $\frac{5}{2}$. Now multiply the improper fractions: $\frac{5}{3} \times \frac{5}{2} = \frac{25}{6}$. To convert this back to a mixed number, divide 25 by 6: $25 ÷ 6 = 4$ remainder $1$, so $\frac{25}{6} = 4\frac{1}{6}$. Choice A ($3\frac{1}{3}$) likely comes from incorrectly adding the whole numbers and fractions separately: $1 + 2 = 3$ and $\frac{2}{3} \times \frac{1}{2} = \frac{1}{3}$. This doesn't work because you can't distribute multiplication over addition this way. Choice B ($4\frac{2}{3}$) might result from calculation errors in the conversion or multiplication steps. Choice C ($3\frac{5}{6}$) could come from multiplying $1 \times 2 = 2$ for the whole number part, then adding $\frac{2}{3} \times \frac{1}{2} = \frac{1}{3}$ incorrectly, or from other computational mistakes. The correct answer is D ($4\frac{1}{6}$). Study tip: Always convert mixed numbers to improper fractions before multiplying. Never try to multiply the parts separately—it leads to wrong answers every time.
6. Nina bought 3 notebooks at $1.45 each and a pen for $0.95. She paid with a $10 bill. How much change should she receive?
   Answer: $4.70 This is a multi-step money problem that tests your ability to calculate total cost and find change. When you see word problems involving purchases and change, break them down into clear steps: find the total cost, then subtract from the amount paid. First, calculate the total cost of Nina's purchases. She bought 3 notebooks at $1.45 each, so the notebooks cost $3 \times \$1.45 = \$4.35$. Adding the pen for $0.95, her total purchase is $\$4.35 + \$0.95 = \$5.30$. Since Nina paid with a $10 bill, her change is $\$10.00 - \$5.30 = \$4.70$. This confirms that A is correct. Let's examine why the other answers are wrong. Answer B ($4.30) likely comes from miscalculating the notebooks as costing $4.30 total instead of $4.35, then adding the pen incorrectly. Answer C ($5.70) represents the total cost plus an error—perhaps someone added $0.40 extra somewhere in their calculations. Answer D ($3.65) could result from forgetting to include one of the notebooks in the total cost calculation. When working money problems on the TACHS, always double-check your arithmetic by working backwards: add your calculated change to the total cost to verify you get the original amount paid. Also, estimate first—since Nina spent about $5.30 out of $10, her change should be close to $4.70, which helps you spot obvious errors quickly.
7. What is the value of $48 \div (6 \times 2) + 7$?
   Answer: 11 This question tests your understanding of order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right). When you see an expression with multiple operations, you must follow the correct sequence. Start with $48 \div (6 \times 2) + 7$. First, handle what's inside the parentheses: $6 \times 2 = 12$. Now you have $48 \div 12 + 7$. Next, perform the division: $48 \div 12 = 4$. Finally, add: $4 + 7 = 11$. Looking at the wrong answers, choice A (10) likely comes from incorrectly calculating $48 \div 12 = 3$ instead of 4, then adding 7. Choice C (15) could result from doing the operations left to right without respecting order of operations: $48 \div 6 = 8$, then $8 \times 2 = 16$, then $16 + 7 = 23$ (though this doesn't match exactly, it shows the confusion from ignoring parentheses). Choice D (19) might come from calculating $48 \div 6 = 8$, then $8 \times 2 + 7 = 23$ and making an arithmetic error, or from $6 \times 2 + 7 = 19$ while completely ignoring the division. The correct answer is B (11). Remember: When you see parentheses in an expression, they're your first priority. Always solve what's inside them before moving to other operations. This prevents the most common mistakes on order of operations problems.
8. Evaluate $\dfrac{9}{10} \div \dfrac{3}{5}.$ Give your answer as a decimal.
   Answer: 1.5 When you encounter division of fractions, remember that dividing by a fraction is the same as multiplying by its reciprocal. This is a fundamental operation that appears frequently on standardized tests. To solve $\frac{9}{10} \div \frac{3}{5}$, you need to multiply $\frac{9}{10}$ by the reciprocal of $\frac{3}{5}$, which is $\frac{5}{3}$. This gives you: $\frac{9}{10} \times \frac{5}{3} = \frac{9 \times 5}{10 \times 3} = \frac{45}{30}$. Simplifying this fraction by dividing both numerator and denominator by 15, you get $\frac{3}{2} = 1.5$. Therefore, answer A is correct. Let's examine why the other options are wrong. Option B (1.2) likely comes from incorrectly multiplying $\frac{9}{10} \times \frac{3}{5} = \frac{27}{50} = 0.54$, then making an arithmetic error. Option C (0.6) results from directly multiplying the fractions without converting to division: $\frac{9}{10} \times \frac{3}{5} = \frac{27}{50} = 0.54$, rounded incorrectly. Option D (0.3) might come from dividing incorrectly, perhaps computing $\frac{3}{10}$ somehow. Remember this key strategy: whenever you see fraction division, immediately think "multiply by the reciprocal." Also, always convert your final fraction to a decimal carefully—improper fractions like $\frac{3}{2}$ will give you numbers greater than 1, which can help you eliminate obviously incorrect decimal answers.
9. Evaluate $\left(\dfrac{5}{6}+0.25\right)-\dfrac{1}{3}.$
   Answer: $\dfrac{3}{4}$ When you encounter mixed operations with fractions and decimals, your first step should be to convert everything to the same format - either all fractions or all decimals - to avoid calculation errors. Let's convert the decimal 0.25 to a fraction: $0.25 = \frac{25}{100} = \frac{1}{4}$. Now we can rewrite the expression as: $\left(\frac{5}{6} + \frac{1}{4}\right) - \frac{1}{3}$ To add $\frac{5}{6} + \frac{1}{4}$, find the least common denominator. The LCM of 6 and 4 is 12: $\frac{5}{6} = \frac{10}{12}$ and $\frac{1}{4} = \frac{3}{12}$ So $\frac{5}{6} + \frac{1}{4} = \frac{10}{12} + \frac{3}{12} = \frac{13}{12}$ Now subtract $\frac{1}{3}$. Convert to twelfths: $\frac{1}{3} = \frac{4}{12}$ Therefore: $\frac{13}{12} - \frac{4}{12} = \frac{9}{12} = \frac{3}{4}$ Choice A ($1$) might result from incorrectly adding all terms instead of following order of operations. Choice B ($\frac{2}{3}$) could come from computational errors when finding common denominators. Choice C ($\frac{5}{6}$) might result from ignoring the other terms entirely or making sign errors. Choice D ($\frac{3}{4}$) is correct. Study tip: Always convert mixed formats to one type before calculating. When working with fractions, find the least common denominator for all terms at once rather than converting step-by-step to minimize errors.
10. Find $5\dfrac{1}{4}-2\dfrac{5}{6}.$
    Answer: $2\dfrac{5}{12}$ When you encounter mixed number subtraction, the key is finding a common denominator for the fractions and borrowing when necessary. First, convert both mixed numbers to have the same denominator. The fractions are $\frac{1}{4}$ and $\frac{5}{6}$. The least common denominator of 4 and 6 is 12. So $5\frac{1}{4} = 5\frac{3}{12}$ and $2\frac{5}{6} = 2\frac{10}{12}$. Now you have $5\frac{3}{12} - 2\frac{10}{12}$. Since $\frac{3}{12} < \frac{10}{12}$, you can't subtract the fractions directly. You need to borrow 1 from the whole number part: $5\frac{3}{12} = 4\frac{15}{12}$ (since $1 = \frac{12}{12}$, so $\frac{3}{12} + \frac{12}{12} = \frac{15}{12}$). Now subtract: $4\frac{15}{12} - 2\frac{10}{12} = 2\frac{5}{12}$. This confirms answer C. Looking at the wrong answers: A) $2\frac{1}{2}$ suggests you might have approximated or made an error with the common denominator. B) $3\frac{1}{12}$ indicates you likely forgot to borrow when the fraction part became negative, or miscalculated the whole number subtraction. D) $3\frac{5}{12}$ shows you got the fraction part right but subtracted $5-2=3$ for the whole numbers without accounting for the borrowing step. Remember: when subtracting mixed numbers, always convert to a common denominator first, then borrow from the whole number if the first fraction is smaller than the second.
11. Nina bought 3 notebooks at $$1.45 each and a pen for $$0.95. She paid with a $$10 bill. How much change should she receive?
    Answer: $$4.70 This problem tests your ability to work through a multi-step money calculation involving purchasing items and making change. When you see questions about buying multiple items and calculating change, break the problem into clear steps: find the total cost, then subtract from the amount paid. First, calculate the total cost of Nina's purchases. She bought 3 notebooks at $1.45 each, so the notebooks cost $3 \times \$1.45 = \$4.35$. Adding the pen at $0.95, her total purchase is $\$4.35 + \$0.95 = \$5.30$. Since Nina paid with a $10 bill, her change is $\$10.00 - \$5.30 = \$4.70$, which is answer D. Let's examine why the other choices are wrong. Choice A ($3.65) would result if you incorrectly calculated the notebook total as $2.45 instead of $4.35, perhaps by only counting 2 notebooks instead of 3. Choice B ($4.30) appears if you forget to add the pen cost entirely, calculating change as $10.00 - $4.35 - $1.45 = $4.30 instead of properly finding the total first. Choice C ($5.70) is the result of subtracting incorrectly, getting $10.00 - $4.30 instead of the correct $5.30 total. For money word problems, always organize your work systematically: list all costs, find the subtotal, add tax if applicable, then calculate change. Double-check your arithmetic at each step, especially when multiplying quantities by unit prices.
12. Find $7.5 \div 12.$
    Answer: 0.625 When you encounter decimal division, you can solve it by converting to fraction form or using long division. Let's work through $7.5 \div 12$ step by step. The most straightforward approach is to perform long division. Set up $7.5 \div 12$ as $7.500 \div 12$ (adding zeros as needed). Since 7.5 is less than 12, your answer will be less than 1, so start with "0." and continue dividing. $12$ goes into $75$ six times ($6 \times 12 = 72$), leaving a remainder of $3$. Bring down the next zero to get $30$. $12$ goes into $30$ twice ($2 \times 12 = 24$), leaving a remainder of $6$. Bring down another zero to get $60$. $12$ goes into $60$ exactly five times ($5 \times 12 = 60$). This gives you $0.625$. Choice A (0.750) represents what you'd get if you mistakenly calculated $7.5 \div 10$ instead of $7.5 \div 12$. Choice B (0.600) might result from rounding errors or incorrectly placing the decimal point. Choice C (0.650) could come from calculation mistakes in the long division process. Choice D (0.625) is correct. Study tip: For decimal division problems, always double-check your work by multiplying your answer back: $0.625 \times 12 = 7.5$ ✓. This reverse calculation catches most common errors and builds confidence in your division skills.
13. What is $0.34 \times 2.5$?
    Answer: 0.85 When multiplying decimals, you can treat them like whole numbers initially, then place the decimal point correctly in your final answer. First, multiply $34 \times 25 = 850$. To do this: $34 \times 25 = 34 \times 20 + 34 \times 5 = 680 + 170 = 850$. Now count the total decimal places in both original numbers: $0.34$ has 2 decimal places, and $2.5$ has 1 decimal place, giving you 3 total decimal places. Place the decimal point 3 places from the right in your answer: $850$ becomes $0.850$, or simply $0.85$. This confirms answer C is correct. Looking at the wrong answers: Choice A (8.5) represents forgetting to account for decimal places entirely—you'd get this if you treated the problem as $34 \times 2.5 = 85$, then randomly placed one decimal. Choice B (0.085) shows a common error of miscounting decimal places—this would result from placing the decimal 4 places from the right instead of 3. Choice D (0.75) appears to come from incorrect multiplication, possibly confusing this with $0.3 \times 2.5$ or making an arithmetic error in the basic multiplication step. Remember this decimal multiplication pattern: multiply the numbers as if they're whole numbers, then count the total decimal places in both factors to determine where your decimal point goes in the final answer. This systematic approach prevents the placement errors that create most wrong answers on decimal multiplication questions.

Start this quiz