Understand Positive and Negative Numbers
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6th Grade Math › Understand Positive and Negative Numbers
A stock's price changes over five days were: $$+\3.50$$, $$-\2.25$$, $$+\1.75$$, $$-\4.00$$, $$+\0.50$$. If the stock started at $$\25.00$$ per share, what does it mean when a stock price change is negative versus positive?
Negative changes mean the company paid dividends to shareholders; positive changes show stock splits occurred; final price is $$\30.50$$.
Negative changes mean the stock lost value that day; positive changes mean it gained value; final price is $$\24.50$$.
Negative changes indicate weekend days when markets are closed; positive changes show weekday trading; final price is $$\25.00$$.
Negative changes mean more people sold than bought; positive changes mean trading volume increased; final price is $$\24.50$$.
Explanation
Price changes: 25.00 + 3.50 - 2.25 + 1.75 - 4.00 + 0.50 = 24.50. Positive changes indicate the stock gained value (price increased), while negative changes indicate the stock lost value (price decreased). Zero change would mean the price stayed the same. Choice B has correct final price but wrong interpretation of positive/negative. Choice C misunderstands stock market mechanics. Choice D has wrong price calculation and wrong interpretation.
A student starts with $\$0$ in a lunch account. Deposits are positive and money owed is negative. After buying lunch, the account shows $-12$. What does $-12$ mean in this context?
The student spent $\$12$ but still has $$0$.
The student owes $\$12$.
The student has exactly $\$12$ and no more money can be added.
The student has $\$12$ saved.
Explanation
This question tests understanding that positive and negative numbers describe opposite values, such as credits or debits in money, representing quantities in financial contexts, and explaining zero's meaning as a balanced account with no debt or surplus. Positive numbers represent deposits or money added, like +$50 meaning $50 credited, while negative numbers represent money owed, like -$30 meaning $30 debt; in temperature, +15°C is above freezing and -8°C is below; in elevation, +450 m is above sea level and -230 m is below, with zero as the reference like $0 for balance, 0°C for freezing, or 0 m for sea level. For example, +$50 means $50 credited or saved, -$30 means $30 owed or debt, and $0 means exactly balanced, neither owing nor having extra. The correct meaning of -12 is that the student owes $12, as the negative sign indicates a debt below the zero balance. A common error is interpreting negative as having money saved, like positive, or thinking it means spent but back to zero, or confusing zero as a limit instead of neutral balance. To represent money in accounts, identify if it's credit (positive) or debt (negative), assign the sign like -12 for owing 12, and include units $; zero means balanced, not nothing or a cap. Comparing negatives, -$5 is better than -$10 because it's less debt, closer to zero, and signed numbers extend to negatives for full financial representation, avoiding mistakes like reversed signs or unclear zero.
A scientist measures electric charge on an object as it gains and loses electrons: starts with $$0$$ charge, gains $$15$$ electrons ($$-15$$ charge units), loses $$8$$ electrons ($$+8$$ charge units), gains $$12$$ electrons ($$-12$$ charge units). What does zero charge represent, and what is the object's final charge?
Zero represents the starting point before any measurements; final charge is $$+35$$ units from the total electron activity.
Zero represents when the object cannot conduct electricity; final charge is $$-35$$ units from adding all electron movements.
Zero represents the maximum charge an object can safely hold; final charge is $$+19$$ units indicating excess protons.
Zero represents equal numbers of protons and electrons with no net charge; final charge is $$-19$$ units indicating excess electrons.
Explanation
When you encounter electric charge problems, remember that charge is all about the balance between positively charged protons and negatively charged electrons. Zero charge means an object has equal numbers of protons and electrons, creating no net electrical effect.
Let's track the changes step by step. Starting from zero charge (balanced protons and electrons), the object gains 15 electrons, giving it $$-15$$ charge units. Then it loses 8 electrons, which adds $$+8$$ charge units (losing negative charge makes it more positive). Finally, it gains 12 more electrons, adding $$-12$$ charge units.
Calculate the final charge: $$0 + (-15) + (+8) + (-12) = -19$$ charge units. This negative result means the object has 19 more electrons than protons.
Choice A incorrectly suggests zero is just a starting measurement point and miscalculates the final charge as positive. Choice B wrongly defines zero as a safety maximum and gets both the sign and magnitude wrong. Choice C misunderstands zero as related to electrical conductivity and incorrectly adds the absolute values of all changes ($$15 + 8 + 12 = 35$$) instead of tracking the net effect.
Choice D correctly identifies that zero charge represents balanced protons and electrons, and accurately calculates the final charge as $$-19$$ units, indicating excess electrons.
Study tip: In charge problems, always track the running total as you add and subtract electrons. Remember that gaining electrons makes charge more negative, while losing electrons makes it more positive.
A football team's progress during four plays shows: $$+12$$ yards, $$-8$$ yards, $$-15$$ yards, and $$+6$$ yards. If they started at their own 35-yard line, what does zero represent on a football field, and what would a negative position mean?
Zero represents midfield; a negative position would mean they moved backward past their starting point into worse field position.
Zero represents their goal line; a negative position would mean they moved backward into their own end zone.
Zero represents their goal line; a negative position would mean the opposing team scored a safety against them.
Zero represents the opposing team's goal line; a negative position would mean they gained yards beyond the touchdown line.
Explanation
Net progress: +12 - 8 - 15 + 6 = -5 yards from the 35-yard line, putting them at the 30-yard line. In football field position context, zero typically represents a goal line. A negative position would mean moving backward past the goal line into the end zone (potentially a safety). Choice A incorrectly identifies midfield as zero. Choice B has correct zero meaning but wrong consequence. Choice D reverses the field orientation.
An elevator starts at the ground floor (level 0) and makes these moves: up $$8$$ floors, down $$12$$ floors, up $$3$$ floors, down $$7$$ floors. If basement levels are numbered with negative integers, what is the elevator's final position and what situation would result in a zero reading?
The elevator is at level $$+8$$, and zero occurs when the elevator has made an equal number of up and down trips.
The elevator is at level $$-8$$, and zero occurs when the elevator is exactly between the ground floor and first basement.
The elevator is at level $$-30$$, and zero occurs when the elevator stops moving between floors for safety reasons.
The elevator is at level $$-8$$, and zero occurs when the elevator returns to the ground floor reference point.
Explanation
Starting at 0: 0 + 8 - 12 + 3 - 7 = -8 (8th basement level). Zero represents the ground floor, the natural reference point between above-ground positive floors and below-ground negative floors. A zero reading occurs when returning to this reference level. Choice A has correct position but wrong meaning of zero. Choice C has wrong calculation. Choice D has completely wrong calculation and meaning.
A mountain climber's elevation changes during a day: starts at base camp ($$5,000$$ feet above sea level), climbs up $$2,800$$ feet, descends $$1,200$$ feet, climbs up $$900$$ feet, then descends $$3,100$$ feet. If sea level represents zero elevation, what would a negative elevation indicate?
A negative elevation would indicate nighttime climbing hours; the climber ends at $$1,100$$ feet above sea level.
A negative elevation would indicate dangerous climbing conditions; the climber ends at $$4,400$$ feet above sea level.
A negative elevation would indicate being underground in caves; the climber ends at $$12,000$$ feet above sea level.
A negative elevation would indicate being below sea level, like Death Valley; the climber ends at $$4,400$$ feet above sea level.
Explanation
When working with elevation problems, you're dealing with positive and negative numbers on a number line where sea level is zero. Positive elevations are above sea level, and negative elevations are below sea level.
Let's track the climber's position step by step. Starting at base camp: $$5,000$$ feet. After climbing up $$2,800$$ feet: $$5,000 + 2,800 = 7,800$$ feet. After descending $$1,200$$ feet: $$7,800 - 1,200 = 6,600$$ feet. After climbing up $$900$$ feet: $$6,600 + 900 = 7,500$$ feet. Finally, after descending $$3,100$$ feet: $$7,500 - 3,100 = 4,400$$ feet above sea level.
Now for what negative elevation means: since sea level is zero, any negative elevation indicates being below sea level. This happens in real places like Death Valley, which sits $$282$$ feet below sea level.
Answer A incorrectly suggests negative elevation means being in underground caves and gives the wrong final elevation of $$12,000$$ feet. Answer B incorrectly connects negative elevation to dangerous conditions and gives the right final elevation but wrong reasoning. Answer C incorrectly links negative elevation to nighttime climbing and miscalculates the final elevation as $$1,100$$ feet.
Answer D correctly explains that negative elevation means below sea level (like Death Valley) and correctly calculates the final elevation as $$4,400$$ feet above sea level.
Remember: in elevation problems, always use sea level as your zero reference point. Negative elevations are real geographical features, not abstract concepts.
A company's quarterly profits and losses were: Q1 gained $$\50,000$$, Q2 lost $$\75,000$$, Q3 gained $$\40,000$$, Q4 lost $$\30,000$$. Starting from a baseline of $$\0$$, what does the final negative result indicate about the company's annual performance?
The company had a net gain of $$\15,000$$ for the year, meaning their profitable quarters outweighed their losses.
The company broke exactly even at $$\0$$ because they had both profits and losses during the year.
The company had a net loss of $$\195,000$$ by adding all quarterly amounts together without considering signs.
The company had a net loss of $$\15,000$$ for the year, meaning they spent more than they earned overall.
Explanation
Net result: +50,000 - 75,000 + 40,000 - 30,000 = -15,000. The negative result indicates the company spent $15,000 more than it earned, representing an overall loss. Zero would represent breaking even (total income equals total expenses). Choice B gives wrong sign for the result. Choice C incorrectly treats all amounts as positive. Choice D incorrectly states they broke even.
A submarine starts at sea level and descends $$150$$ feet. It then rises $$75$$ feet and descends another $$90$$ feet. If sea level represents $$0$$ feet, which statement best describes the submarine's final position and what zero represents in this context?
The submarine is at $$+165$$ feet, and zero represents the deepest point the submarine can safely reach.
The submarine is at $$-315$$ feet, and zero represents sea level as the reference point between above and below water.
The submarine is at $$-165$$ feet, and zero represents sea level as the reference point between above and below water.
The submarine is at $$-165$$ feet, and zero represents the ocean floor where depth measurements begin.
Explanation
Starting at 0, the submarine moves: 0 - 150 + 75 - 90 = -165 feet. Zero represents sea level, the natural reference point that separates above-water (positive) from below-water (negative) positions. Choice A has the correct position but wrong meaning of zero. Choice C incorrectly adds all movements as negative. Choice D has both wrong position and wrong meaning of zero.
A robot starts at position $0$ on a number line. Moving forward is positive and moving backward is negative. Which signed number represents moving 9 steps backward from the start?
$0$
$-9$
$+9$
$9$
Explanation
This question tests understanding that positive and negative numbers describe opposite directions or values, such as forward or backward movement on a number line, representing quantities in contexts like robotics, and explaining zero's meaning as the starting point. Positive numbers represent forward steps, like +9 meaning 9 steps ahead, while negative represent backward, like -9 meaning 9 steps back; in temperature, +15°C is above and -8°C below freezing; in money, +$50 is credit and -$30 is debt, with zero as reference like 0 for start, 0°C for freezing, or $0 for balance. For example, +9 is 9 steps forward from start, -9 is 9 steps backward, and 0 is remaining at the starting position. In this case, -9 correctly represents moving 9 steps backward, as the negative sign shows the opposite direction. A common error is omitting the sign or thinking backward is positive, or seeing zero as nothing when it's the specific start, or reversing comparisons like -9 > -5 when -5 > -9. To represent movements, identify forward (positive) or backward (negative), assign the sign like -9 for backward, without units here; zero means the starting point. Understanding signed numbers extends the line to negatives, but mistakes include wrong sign or comparison reversal.
A game uses a score change system: gaining points is positive and losing points is negative. A player’s score change is $+9$. Which situation matches $+9$?
The player gained 9 points.
The player has 9 points total and cannot change.
The player’s score is 0 points.
The player lost 9 points.
Explanation
This question tests understanding that positive and negative numbers describe opposite values, such as gaining or losing points in a game, representing quantities in scoring contexts, and explaining zero's meaning as no change in score. Positive numbers represent gains, like +9 meaning 9 points added, negative losses, like -9 meaning 9 points subtracted; in temperature, +15°C above and -8°C below; in elevation, +450 m above and -230 m below, with zero as no change, freezing, or sea level. For example, +9 is gain of 9, -9 is loss of 9, 0 is no score change. The correct situation for +9 is the player gained 9 points, as positive indicates addition. A common error is thinking positive means loss or total fixed at 9, or confusing with zero score. To represent score changes, identify gain (positive) or loss (negative), assign sign like +9 for gain; zero means balanced no change. Comparing negatives, -5 > -10 as smaller loss; signed numbers extend for both, avoiding sign reversal mistakes.