QuanticaPrealgebraOpen in the courseThe last three lessons gave you three tools, reordering and pairing sums, handling signs, and multiplying with the distributive bridge. Now they get mixed together. The problems below carry no labels, so nothing tells you which tool a problem needs, and several need two at once. Real problems never tell you which chapter they came from. Read slowly, look for structure first, and expect every one of these to come apart without paper.
Problem
For 14 days, a bakery celebrates its birthday. On day 1 it gives away 1 free roll, on day 2 it gives 2, and so on up to 14 rolls on day 14, and it runs the same giveaway at each of its $5$ branches. How many rolls does the whole chain give away?
Show a hint
- First find one branch’s total: pair \(1 + 14\), \(2 + 13\), …, the trick you learned from a seven-year-old.
Show the full solution
Pair the days, \(1 + 14, 2 + 13, \ldots, 7 + 8\), into seven 15s, so \(7 \times 15 = 105\) rolls per branch. Then \(5 \times 105 = \boxed{525}\).
Problem
Zoe rings up six items on Monday costing $41, 52, 63, 74, 85, 96$. On Tuesday she rings up $39, 50, 61, 72, 83, 94$. Without finding either total, work out how much more Monday’s bill is than Tuesday’s.
Show the full solution
Match the lists item by item. Every Monday item costs 2 more, and there are six items, so Monday’s bill is \(6 \times 2 = \boxed{12}\) more. Neither total was needed.
Problem
A freight scale reports that crate A outweighs crate B by $9$ kg, that is, \(A - B = 9\). The mover wants the comparison the other way around. What is \(B - A\)?
Show the full solution
\(B - A = -(A - B) = \boxed{-9}\). Reversing a subtraction gives the opposite of the original difference.
Problem
Here is a puzzle. Split the numbers $3, 5, 8, 0$ into two pairs, multiply inside each pair, then add the two results. What is the largest total you can reach?
Show a hint
- Whatever number 0 is paired with vanishes from the final total. Which number can you best afford to lose?
Show the full solution
Pair 0 with 3, so the other pair is \(5 \times 8 = 40\) and the total is \(40 + 0 = \boxed{40}\). Anything paired with 0 drops out of the total, so spend the 0 on the smallest number. The other pairings give only 15 and 24.
Problem
Compute $$(445 - (4 \times 111)) \times 777.$$
Show a hint
- Work out the inner parentheses first: what is \(4 \times 111\)?
Show the full solution
Inside, \(4 \times 111 = 444\), so the parentheses hold \(445 - 444 = 1\), and the whole expression is \(1 \times 777 = \boxed{777}\). A big expression can hide a factor of 1.
Problem
Twins Ria and Kio race through a worksheet. Ria computes \(46 \times 45\), and Kio computes \(46 \times 44 + 46\). Their teacher asks for Ria’s answer minus Kio’s. Find it the lazy way.
Show the full solution
Kio’s \(46 \times 44 + 46\) is forty-four 46s plus one more, which is forty-five 46s, exactly Ria’s \(46 \times 45\). The two are equal, so their difference is \(\boxed{0}\).
Problem
A frog sits at 0 on a number line. It makes $8$ jumps of \(+5\), then $6$ jumps of \(-7\). Where does the frog land?
Show the full solution
The jumps are two products, \(8 \times 5 = 40\) forward, then \(6 \times (-7) = -42\) back. The frog lands at \(40 + (-42) = \boxed{-2}\).
Problem
Compute $$18 \times 7 + 18 \times 21 + 18 \times 12.$$
Show the full solution
All three products share the 18, so factor it out. \(18 \times (7 + 21 + 12) = 18 \times 40 = \boxed{720}\).
Problem
Compute $$(294 + 295 + 296 + \cdots + 306) - (13 \times 294).$$
Show a hint
- Every number in the run is 294 carrying a little extra, \(294 + 0\), \(294 + 1\), …, \(294 + 12\).
- Subtracting \(13 \times 294\) strips one 294 out of every term. What extras are left behind?
Show the full solution
Each number from 294 to 306 is 294 plus a small excess, \(294 + 0, 294 + 1, \ldots, 294 + 12\). Subtracting \(13 \times 294\) strips the 294 from every term, leaving \(1 + 2 + \cdots + 12\), which pairs into six 13s for \(\boxed{78}\).
Problem
Two siblings argue. Asha says \(399 \times 501\) and \(400 \times 500\) must be equal, “because you just move a 1 from one number to the other.” Settle it. Compute the larger minus the smaller.
Show a hint
- Both products have \(399 \times 500\) hiding inside.
- \(399 \times 501 = 399 \times 500 + 399\), while \(400 \times 500 = 399 \times 500 + 500\).
Show the full solution
Peel a layer off each, \(399 \times 501 = 399 \times 500 + 399\) and \(400 \times 500 = 399 \times 500 + 500\). They share the same bulk, so \(400 \times 500\) is larger by \(500 - 399 = \boxed{101}\).
Problem
Compute $$64 \times 36 - 63 \times 35$$ without long multiplication. (Careful, no factor is shared… yet.)
Show a hint
- Manufacture a shared factor: rewrite 64 as \(63 + 1\).
- Then \(64 \times 36 = 63 \times 36 + 36\), and now two of the products share a 63.
- \(63 \times 36 - 63 \times 35\) collapses. Add back the spare 36.
Show the full solution
Write \(64 \times 36 = (63 + 1) \times 36 = 63 \times 36 + 36\). Then \(63 \times 36 + 36 - 63 \times 35 = 63 \times (36 - 35) + 36 = 63 + 36 = \boxed{99}\). When no shared factor exists, build one.
If any of those three still feels wobbly, revisit its lesson before moving on. Up next is the last of the four operations, division, and it will fall to the same playbook.