An empty box and a letter can both represent a number, but their roles come from the surrounding statement. An equality specifies a relationship the number must satisfy. A cost expression describes how the total depends on a quantity. Begin by naming the unknown before manipulating its symbol.
In these examples, the numbers and prices are fictional conditions. You are checking a mathematical relationship, rather than comparing current shop prices. Keep a distinction between a number to be found and a letter that represents different possible inputs. Both uses can be checked by substitution.
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An equality can be checked in both directions
For □ + 18 = 45, subtract 18 from 45 to obtain 27. Then replace the box with 27 and check that 27 + 18 is 45. This second reading tests the original relationship instead of only repeating the subtraction.
For x pencils at 80 yen each and one eraser at 100 yen, the total is 80x + 100 yen. The multiplication belongs to the number of pencils; the fixed eraser price is added once. Explain which part of the purchase each term describes.
Find the multiplier from corresponding pairs
The supplied pairs x = 1, 2, 3 and y = 4, 8, 12 satisfy y = kx with k = 4. Divide each y value by its paired nonzero x value to check the same multiplier. Do not mix a value from one row with a value from another.
The question supplies the proportional form y = kx. Matching these three pairs verifies the requested k under that form; it does not prove that any possible table must continue proportionally. Retain the stated rule when explaining what you have established.
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Check your understanding with examples
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Per-ID original mathematical derivations and faithful English adaptations for 300 existing families, plus original fictional data and diagram conditions. Supports mathematical reasoning, not external facts or independent review approval.
Source checked: 2026-10-04 · Original authoring and derivation record
Example 2 · Algebra and functions
Each pencil costs 80 yen. You buy x pencils and one eraser costing 100 yen. What is the total cost?
(80x + 100) yen
180x yen
(80 + 100x) yen
(80x − 100) yen
Read the answer and explanation
Answer: (80x + 100) yen
The pencils cost 80x yen and the eraser adds a fixed 100 yen, so the total is 80x + 100 yen.
Per-ID original mathematical derivations and faithful English adaptations for 300 existing families, plus original fictional data and diagram conditions. Supports mathematical reasoning, not external facts or independent review approval.
Source checked: 2026-10-04 · Original authoring and derivation record
Example 3 · Algebra and functions
When x is 1, 2 and 3, y is 4, 8 and 12 respectively. If y = kx, what is k?
2
12
4
3
Read the answer and explanation
Answer: 4
Divide y by x for each pair: 4/1 = 8/2 = 12/3 = 4. This common ratio is k.
Per-ID original mathematical derivations and faithful English adaptations for 300 existing families, plus original fictional data and diagram conditions. Supports mathematical reasoning, not external facts or independent review approval.
Source checked: 2026-10-04 · Original authoring and derivation record
Topics: Algebra and functions. Range: Everyday knowledge. Difficulty: Basic, Standard. These are selected initially. You can change these on the setup screen.
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