QuanticaPrealgebraOpen in the courseThe last three lessons gave you three tools. You can reorder and pair sums, handle signs, and multiply using the distributive property. The problems below are unlabelled, so nothing tells you which tool a problem needs, and several need two at once. Read slowly, look for structure first, and expect to solve each one 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 branches. How many rolls does the whole chain give away?
Show a hint
- First find one branch’s total: pair , , …, the trick you learned from a seven-year-old.
Show the full solution
Pair the days, , into seven 15s, so rolls per branch. Then .
Problem
Zoe rings up six items on Monday costing . On Tuesday she rings up . 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 more. Neither total was needed.
Problem
A freight scale reports that crate A outweighs crate B by kg, that is, . The mover wants the comparison the other way around. What is ?
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. Reversing a subtraction gives the opposite of the original difference.
Problem
Here is a puzzle. Split the numbers 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?
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Pair 0 with 3, so the other pair is and the total is . 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
Show a hint
- Work out the inner parentheses first: what is ?
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Inside, , so the parentheses hold , and the whole expression is . A big expression can hide a factor of 1.
Problem
Twins Ria and Kio race through a worksheet. Ria computes , and Kio computes . Their teacher asks for Ria’s answer minus Kio’s. Find it the lazy way.
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Kio’s is forty-four 46s plus one more, which is forty-five 46s, exactly Ria’s . The two are equal, so their difference is .
Problem
A frog sits at 0 on a number line. It makes jumps of , then jumps of . Where does the frog land?
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The jumps are two products, forward, then back. The frog lands at .
Problem
Compute
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All three products share the 18, so factor it out. .
Problem
Compute
Show a hint
- Every number in the run is 294 carrying a little extra, , , …, .
- Subtracting strips one 294 out of every term. What extras are left behind?
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Each number from 294 to 306 is 294 plus a small excess, . Subtracting strips the 294 from every term, leaving , which pairs into six 13s for .
Problem
Two siblings argue. Asha says and 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 hiding inside.
- , while .
Show the full solution
Peel a layer off each, and . They share the same bulk, so is larger by .
Problem
Compute without long multiplication. (Careful, no factor is shared… yet.)
Show a hint
- Manufacture a shared factor: rewrite 64 as .
- Then , and now two of the products share a 63.
- collapses. Add back the spare 36.
Show the full solution
Write . Then . 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.