Depth · one concept from first principles · step 2 of 4
A natural number is less than . When is written as a decimal, its tenths digit is and its thousandths digit is . When is written as a decimal, its hundredths digit is . Find .
Since n ＜ 75, n/75 ＜ 1, so a tenths digit of 1 means 0.1 ≤ n/75 ＜ 0.2. Multiply by 75.
So n is one of 8, 9, …, 14.
7.EE.B.4Convert To AlgebraThe tenths digit of is exactly when , that is, .
A number below with tenths digit is plus whatever its later digits add, and those later digits always add up to less than one more tenth.
Ten of any place make one of the place to its left, so even all s after the tenths never fill a whole extra tenth.
Multiplying all three parts by the positive number stretches the number line without turning it, so .
(n+1)/75 is between 0.12 and 0.2. At 0.2 the hundredths digit would be 0, so it is below 0.2 and its tenths digit is 1.
A tenths digit of 2 with hundredths 8 would need 0.28, far too big.
7.NS.A.2Eliminate PossibilitiesTenths digit 1 and hundredths digit 8 mean 0.18 ≤ (n+1)/75 ＜ 0.19. Multiply by 75.
The only whole number in that short range is 14, so n=13.
7.EE.B.4Convert To AlgebraDivide to confirm the clue we have not used yet.
The tenths digit is 1 and the thousandths digit is 3, as required.
7.NS.A.2Guess And CheckA single digit of a decimal is really a range of values -- and ranges are what inequalities are for.
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