Adding & Subtracting Rational Numbers - Pre Algebra

1. A recipe calls for $2\frac{3}{4}$ cups of flour, but Sarah only has $1\frac{5}{8}$ cups. Later, she borrows $\frac{7}{8}$ cups from a neighbor and buys $1\frac{1}{2}$ more cups at the store. After making the recipe, how much flour does she have left?
   Answer: $1\frac{1}{4}$ cups This is a multi-step word problem involving mixed numbers and fractions. You need to track Sarah's flour through several transactions: what she starts with, what she gains, what she uses, and what remains. First, let's find Sarah's total flour before making the recipe. She starts with $1\frac{5}{8}$ cups, borrows $\frac{7}{8}$ cups, and buys $1\frac{1}{2}$ cups. Converting to improper fractions with denominator 8: $1\frac{5}{8} = \frac{13}{8}$, $\frac{7}{8} = \frac{7}{8}$, and $1\frac{1}{2} = \frac{12}{8}$. Adding these: $\frac{13}{8} + \frac{7}{8} + \frac{12}{8} = \frac{32}{8} = 4$ cups total. Next, she uses $2\frac{3}{4}$ cups for the recipe. Converting: $2\frac{3}{4} = \frac{22}{8}$. Her remaining flour is $4 - 2\frac{3}{4} = \frac{32}{8} - \frac{22}{8} = \frac{10}{8} = 1\frac{1}{4}$ cups. Choice A ($1\frac{7}{8}$) likely comes from miscalculating the total flour Sarah had. Choice B ($\frac{7}{8}$) might result from forgetting to add one of her flour sources. Choice C ($\frac{1}{4}$) could happen if you converted $1\frac{1}{4}$ incorrectly or subtracted the wrong amount. The correct answer is D: $1\frac{1}{4}$ cups. Study tip: In multi-step fraction problems, convert all mixed numbers to the same denominator early on. This prevents arithmetic errors and makes addition and subtraction much cleaner. Always double-check your conversions between mixed numbers and improper fractions.
2. Marcus has a checking account with overdraft protection that charges $25 when the balance goes below zero. His account starts with $156.43. He writes checks for $89.17, $45.82, and $38.76 in that order. After the overdraft fee (if any), he deposits $75.50. What is his final balance?
   Answer: $33.18 Starting balance: $156.43. After first check: $156.43 - $89.17 = $67.26. After second check: $67.26 - $45.82 = $21.44. After third check: $21.44 - $38.76 = -$17.32 (overdraft occurs). Overdraft fee: -$17.32 - $25.00 = -$42.32. After deposit: -$42.32 + $75.50 = $33.18. Choice A ignores the overdraft fee. Choice C incorrectly calculates when the overdraft occurs. Choice D forgets to add the final deposit.
3. The temperature at 6 AM was $-8.5°F$. By noon, it had risen $12.7°F$. In the afternoon, it dropped $15.3°F$, and then in the evening it rose again by $6.8°F$. What was the final temperature?
   Answer: $-4.3°F$ Starting at -8.5°F, add 12.7°F: -8.5 + 12.7 = 4.2°F at noon. Then subtract 15.3°F: 4.2 - 15.3 = -11.1°F in the afternoon. Finally add 6.8°F: -11.1 + 6.8 = -4.3°F. Choice B results from incorrectly treating the initial temperature as positive. Choice C comes from calculation errors in the middle steps. Choice D comes from adding all the changes without considering their signs properly.
4. On a coordinate plane, point $P$ is located at $(-3.5, 2.8)$ and point $Q$ is located at $(1.7, -4.2)$. If point $R$ is positioned such that its $x$-coordinate equals the sum of the $x$-coordinates of $P$ and $Q$, and its $y$-coordinate equals the difference of $P$'s $y$-coordinate minus $Q$'s $y$-coordinate, what are the coordinates of point $R$?
   Answer: $(-1.8, 7.0)$ For point R: x-coordinate = (-3.5) + (1.7) = -1.8. y-coordinate = (2.8) - (-4.2) = 2.8 + 4.2 = 7.0. Therefore R is at (-1.8, 7.0). Choice A uses the wrong sign for the x-coordinate calculation. Choice C incorrectly calculates the y-coordinate as 2.8 - 4.2 = -1.4. Choice D combines the errors from both A and C.
5. A submarine starts at sea level and descends $127\frac{3}{4}$ feet. It then ascends $89\frac{1}{2}$ feet, descends another $45\frac{5}{8}$ feet, and finally ascends $62\frac{7}{8}$ feet. What is the submarine's final depth below sea level?
   Answer: $21\frac{1}{8}$ feet below sea level Let downward be negative and upward be positive, starting at 0. First: 0 - 127¾ = -127¾. Then: -127¾ + 89½ = -127¾ + 89½ = -38¼. Then: -38¼ - 45⅝ = -83⅞. Finally: -83⅞ + 62⅞ = -21⅛. The submarine is 21⅛ feet below sea level. Choice B incorrectly makes the final answer positive. Choice C is the depth after the third movement, before the final ascent. Choice D incorrectly adds all distances regardless of direction.
6. In a card game, positive points are gained and negative points represent losses. Jamie's scores for five rounds were: $+24.5$, $-17.8$, $+9.3$, $-31.6$, and $+18.2$. If players with negative total scores must pay a $5.5$ point penalty, what is Jamie's final score?
   Answer: $2.6$ points This problem tests your ability to work with positive and negative numbers (integers and decimals) and apply conditional rules based on calculation results. Start by finding Jamie's total score from the five rounds. Add all the scores together, being careful with the signs: $+24.5 + (-17.8) + (+9.3) + (-31.6) + (+18.2)$. Group the positive and negative values: $(24.5 + 9.3 + 18.2) + (-17.8 - 31.6) = 52.0 + (-49.4) = +2.6$. Since Jamie's total is $+2.6$ points (positive), she doesn't need to pay the $5.5$ point penalty that only applies to players with negative totals. Therefore, her final score remains $+2.6$ points. Answer A ($8.1$) likely comes from incorrectly adding the penalty instead of subtracting it, or from a calculation error in the initial sum. Answer B ($-2.9$) results from taking the correct total of $+2.6$ and mistakenly subtracting the $5.5$ penalty even though it doesn't apply to positive scores. Answer C ($-8.4$) appears to come from calculation errors in both the initial sum and penalty application. When working with problems involving conditional penalties or bonuses, always calculate the base result first, then check whether the condition is met before applying any additional changes. Pay close attention to positive and negative signs throughout your calculations, and remember that penalties typically only apply when specific conditions are satisfied.
7. A stock's value changed as follows over four days: Day 1: increased by $\frac{3}{8}$, Day 2: decreased by $\frac{5}{6}$, Day 3: increased by $\frac{7}{12}$, Day 4: decreased by $\frac{1}{4}$. If these changes are measured in dollars per share, what is the total change in the stock's value?
   Answer: $-\frac{5}{24}$ dollars per share Sum all changes: +3/8 - 5/6 + 7/12 - 1/4. Convert to common denominator 24: 3/8 = 9/24, 5/6 = 20/24, 7/12 = 14/24, 1/4 = 6/24. Therefore: 9/24 - 20/24 + 14/24 - 6/24 = (9 - 20 + 14 - 6)/24 = -3/24. Since -3/24 simplifies to -1/8, let me recalculate: 9/24 - 20/24 + 14/24 - 6/24 = -5/24. The stock decreased by 5/24 dollars per share.

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