QuanticaPrealgebraOpen in the courseAddition looks too simple to study, but it comes with real freedom. You may add in any order, and you may group the numbers however you like. That freedom is what lets you add fast. Subtraction and negatives turn out to be the same freedom in another form, and the problems below put it to work.
Problem
A vending machine counts coins in the order you drop them. You feed it $8$ coins, then $5$ more, and a friend feeds an identical machine $5$ first, then $8$. How many coins does each machine now hold?
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Both machines hold the same count, \(8 + 5 = 5 + 8 = \boxed{13}\) coins. Swapping the order of a sum never changes the total.
Problem
Add in any order: $$7+8+3.$$
Show a hint
- Pair 7 and 3 to make 10 first!
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Reorder to \(7+3+8\). Then \(7+3=10\), and \(10+8=\boxed{18}\).
Problem
Three jars hold $17$, $3$, and $40$ jellybeans. Maya pours jar 1 into jar 2, then adds jar 3, and Theo pours jar 2 into jar 3, then adds jar 1. How many jellybeans does each end up with?
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Maya gets \((17 + 3) + 40 = 20 + 40 = 60\). Theo gets \(17 + (3 + 40) = 17 + 43 = 60\). Both land on \(\boxed{60}\). Regrouping cannot change a total, so pick the grouping that makes a round number first.
Problem
Now use both freedoms at once: compute $$38 + 47 + 62 + 53$$ in your head.
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Look for numbers that pair to a round total. 38 and 62 make 100, and 47 and 53 make 100.$$(38 + 62) + (47 + 53) = 100 + 100 = \boxed{200}.$$Left to right works too, but every step forces a carry. A quick scan for round-number partners avoids all of them.
Problem
Same idea, a little sneakier. Find $$196 + 297 + 4 + 3$$ with nothing written down.
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196 is 4 short of 200 and 297 is 3 short of 300, so the 4 and the 3 are exactly the missing pieces.$$(196 + 4) + (297 + 3) = 200 + 300 = \boxed{500}.$$When you spot a near-round number, look for the small change that completes it.
Problem
In your head: $$998+997+5+4.$$
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- Round 998 and 997 up to 1000, then take back what you added.
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998 is 2 short of 1000 and 997 is 3 short, and the \(5\) covers both gaps exactly, since \(2+3=5\).$$(998+2)+(997+3)+4 = 1000+1000+4 = \boxed{2004}.$$When a sum mixes near-round numbers with small ones, let the small ones fill the gaps.
Problem
Add the numbers 1 to 10: $$1+2+3+\cdots+10.$$
Show a hint
- Pair 1 with 10, 2 with 9, and each pair makes 11!
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Pair from the ends, \(1+10,\,2+9,\,3+8,\,4+7,\,5+6\), five pairs of \(11\), so \(5\times 11=\boxed{55}\).
Problem
Add every whole number from 1 to 40: $$1 + 2 + 3 + \cdots + 40.$$
Show a hint
- Pair the smallest number with the largest: \(1 + 40\), then \(2 + 39\), then \(3 + 38\), …
- Every pair adds to the same total. What is it, and how many pairs do 40 numbers make?
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Pair from the ends inward.$$(1 + 40) + (2 + 39) + (3 + 38) + \cdots + (20 + 21).$$Each step inward grows one partner by 1 and shrinks the other by 1, so every pair lands on 41. Twenty pairs give$$20 \times 41 = \boxed{820}.$$Grouping neighbours instead gives \(3 + 7 + 11 + \cdots\), all different, so nothing collapses.
Problem
Add every whole number from 1 to 100: $$1+2+\cdots+100.$$
Show a hint
- 50 pairs, each adding to 101.
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Pair from the ends, \(1+100,\,2+99,\,\dots\), fifty pairs, each summing to \(101\). So \(50\times 101 = \boxed{5050}\).
Subtraction is not a new operation, and negatives are not a new headache. Both are addition with a direction attached.
Problem
The temperature reads $8$ degrees, then drops $5$. Every subtraction is really an addition, so rewrite the drop as $$8 + (-5)$$ and compute it.
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Adding \(-5\) means stepping 5 to the left of 8 on the number line, landing on \(\boxed{3}\). Subtracting a number and adding its opposite are the same move. That is \(a-b=a+(-b)\) in action.
Problem
Compute $$10-3-2.$$
Show a hint
- Take away 3, then take away 2.
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Subtracting \(3\) then \(2\) removes \(5\) in all: \(10-5=\boxed{5}\).
Problem
Rearrange cleverly to compute $$50 - 17 + 50 - 33.$$
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Turn every subtraction into adding a negative, then reorder.$$50 + (-17) + 50 + (-33) = (50 + 50) + ((-17) + (-33)) = 100 - 50 = \boxed{50}.$$Minus signs get in the way of rearranging, and rewriting them as additions clears the way.
Problem
Rearrange: $$20-8+20-12.$$
Show a hint
- Look at the two 20s, and what 8 and 12 do together.
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Group: \(20+20=40\) and \(8+12=20\), so \(40-20=\boxed{20}\).
Problem
Compute $$83 - 27 - 33.$$
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Subtract the two together: \(27 + 33 = 60\), so \(83 - 60 = \boxed{23}\).
Problem
Now the signs alternate. Find $$1 - 2 + 3 - 4 + 5 - 6 + \cdots - 12.$$
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- Don’t add left to right. Group the terms into adjacent pairs, \((1 - 2)\), \((3 - 4)\), \((5 - 6)\), …
- What does each pair (an odd number minus the next even number) come out to?
- Each pair is \(-1\), and there are six pairs. So the total is …
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Pair each positive with the negative right after it.$$(1 - 2) + (3 - 4) + (5 - 6) + (7 - 8) + (9 - 10) + (11 - 12)$$Every bracket is \(-1\), and six pairs give$$6 \times (-1) = \boxed{-6}.$$Pairing from the ends inward would give \(-11, 9, -7, 5, -3, 1\), all different, so which pairing collapses changes from problem to problem.
Problem
Find $$100 - 99 + 98 - 97 + \cdots + 2 - 1.$$
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Group in pairs from the left: \((100-99),\,(98-97),\,\ldots,\,(2-1)\) each equal 1. There are 50 pairs, so the total is \(\boxed{50}\).
Problem
Find $$1-2+3-4+\cdots+99-100.$$
Show a hint
- Each pair (1-2),(3-4) makes -1. How many pairs are there?
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Group in pairs: \((1-2)+(3-4)+\cdots+(99-100)\). Each pair is \(-1\), and there are \(50\) pairs, so the total is \(\boxed{-50}\).
One spot trips almost everyone up, the double negative. Why should taking away a negative add? Picture the number line. The opposite of a number is its mirror image across zero. The opposite of 5 is −5, and the opposite of −5 is 5. A minus sign in front means “take the opposite.” So two minus signs in a row, −(−5), say “the opposite of the opposite of 5”, and that lands you right back where you started, at 5.
Problem
A diver is 30 m below sea level, so her depth is written $-30$ m. A drone hovers at exactly the opposite height. Where is the drone? That is, what is $-(-30)$?
Show a hint
- “Opposite” means the mirror image across zero, the same distance from zero but on the other side.
- The opposite of \(-30\) is \(+30\). And \(-(-30)\) is read “the opposite of \(-30\).”
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Opposite means the same distance on the other side of zero, so the opposite of 30 below is 30 above.$$-(-30) = \boxed{30}.$$Each minus sign is one flip across zero, and two flips return home.
Problem
What is $$-(-7)$$?
Show a hint
- The opposite of the opposite is the original.
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The opposite of \(-7\) is \(7\), so \(-(-7)=\boxed{7}\).
Problem
Compute $$3-(-4).$$
Show a hint
- Subtracting a negative is the same as adding.
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Subtracting \(-4\) is the same as adding \(4\): \(3+4=\boxed{7}\).
Problem
Subtracting a negative adds. Convince yourself once, carefully: $$5 - (-6).$$
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The opposite of \(-6\) is \(6\), so taking away \(-6\) is the same as adding \(6\): \(5 + 6 = \boxed{11}\).
Problem
Now untangle a nested one. Work from the inside out. $$7 - (-(-2)).$$
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Work from the inside out. \(-(-2)\) is the opposite of \(-2\), which is 2, so$$7 - (-(-2)) = 7 - 2 = \boxed{5}.$$An even number of minus signs wrapped around a number leaves it positive.
Problem
Last one. Hunt for what cancels: $$12 - (-8) + (-12) - 8.$$
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Every subtraction becomes an addition, since subtracting \(-8\) adds 8 and subtracting 8 adds \(-8\).$$12 + 8 + (-12) + (-8)$$Now 12 sits with \(-12\) and 8 sits with \(-8\), so everything cancels and the sum is \(\boxed{0}\). When positives and negatives mix, check for opposites before adding anything.
Problem
Simplify $$-(-(-(-9))).$$
Show a hint
- Count the minus signs, and pairs cancel.
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Four minus signs, and they cancel in pairs since \(-(-x)=x\), leaving \(\boxed{9}\).
Problem
Hunt for what cancels: $$30 - (-20) - (-10) - 60.$$
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Rewrite the subtractions of negatives as additions: \(30 + 20 + 10 - 60\). The positives make 60, cancelled by the \(-60\), leaving \(\boxed{0}\).
These last problems mix everything from the lesson. Use the full toolkit. Reorder, regroup, turn subtractions into additions, and let opposites cancel.
Problem
Rearrange before you compute: $$45 - 18 + 55 - 82.$$
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Dissolve the subtractions into additions and regroup: \((45 + 55) + ((-18) + (-82)) = 100 - 100 = \boxed{0}\). Rearranging works on any expression once every subtraction becomes adding the opposite.
Problem
Find $$15-(-10)+(-15)-10.$$
Show a hint
- Hunt for what cancels: the 15s and the 10s.
Show the full solution
Rewrite as \(15+10-15-10\). The \(15\)s cancel and the \(10\)s cancel, giving \(\boxed{0}\).
Practice these ideas
Practice
Compute $$28 + 28 + 28 + 72 + 72 + 72.$$
Show the solution
Pair each 28 with a 72: \(28 + 72 = 100\), and there are three such pairs, so the total is \(3 \times 100 = \boxed{300}\).
Practice
Compute $$4998 + 5002 + 4998 + 5002 + 4998 + 5002 + 4998 + 5002.$$
Show the solution
Each neighbouring pair \(4998 + 5002 = 10000\), and there are four pairs, so the sum is \(4 \times 10000 = \boxed{40000}\).
Practice
Compute $$(6 + 16 + 26 + 36) + (4 + 14 + 24 + 34).$$
Show the solution
Pair the two lists term by term into tens: \((6 + 4) + (16 + 14) + (26 + 24) + (36 + 34) = 10 + 30 + 50 + 70 = \boxed{160}\).
Practice
Compute $$64 - 28 + 36 - 22$$ by grouping cleverly.
Show the solution
Gather the numbers that round off: \((64 + 36) - (28 + 22) = 100 - 50 = \boxed{50}\).
Practice
Compute $$88 - 19 - 88 + 44$$, looking for what cancels.
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The 88 and the \(-88\) cancel, leaving \(-19 + 44 = \boxed{25}\).
Practice
Compute $$30 - (-15) - 20 - (-5).$$
Show the solution
Subtracting a negative adds its opposite: \(30 + 15 - 20 + 5 = \boxed{30}\).
Practice
Compute $$-(-9).$$
Show the solution
The opposite of the opposite of 9 brings you back to the start: \(-(-9) = \boxed{9}\).
Practice
Compute $$6 - (-4).$$
Show the solution
Subtracting a negative adds: \(6 - (-4) = 6 + 4 = \boxed{10}\).
Practice
Compute $$-(-(-7))$$, and count the minus signs carefully.
Show the solution
Three minus signs means three flips across zero, \(7 \to -7 \to 7 \to -7\), so \(-(-(-7)) = \boxed{-7}\). An odd number of minus signs leaves a number negative, and an even number brings it back to positive.
Practice
Compute $$(1 + 2 + 3 + \cdots + 40) + (79 + 78 + 77 + \cdots + 41 + 40).$$
Show the solution
The first run is 1–40 and the second is 40–79, so together they list every whole number from 1 to 79, with 40 appearing twice. That is \((1 + 2 + \cdots + 79) + 40 = 3160 + 40 = \boxed{3200}\).
Practice
Around 1785, a German schoolteacher told his class to add up every number from $1$ to $100$, to find $1+2+3+\cdots+100$, pure busywork meant to buy him some quiet. Seconds later, a seven-year-old set down his slate with the answer: $5050$. He had paired $1+100$, $2+99$, $3+98$, $\ldots$, fifty pairs of $101$, and computed $50 \times 101 = 5050$. That boy, Carl Friedrich Gauss, grew up to become one of the greatest mathematicians who ever lived. The trick you just practised is his trick, so you are in very good company. Keep going. 🚀