Number Play | IT

Question 11

Among the numbers 1–100, how many times will the digit ‘7’ occur? Among the numbers 1–1000, how many times will the digit ‘7’ occur?

Solution

Understand the Question

To count the number of times a specific digit (like $7$ ) appears in a range of numbers, count its occurrences across each place value (ones, tens, and hundreds) independently and add them together.
Note the difference between the number of values containing the digit and the total occurrences of the digit: numbers like $77$ or $777$ contain the digit multiple times and contribute to multiple place values.

(i) Among the numbers 1–100, how many times will the digit ‘7’ occur?

Step 1 · Count occurrences of '7' across place values from 1 to 100

Diagram 1

1. In the ones place: Numbers ending in $7$ : $7, 17, 27, 37, 47, 57, 67, 77, 87, 97$ Number of occurrences $= 10$

2. In the tens place: Numbers with $7$ in the tens place: $70, 71, 72, 73, 74, 75, 76, 77, 78, 79$ Number of occurrences $= 10$

\begin{aligned} \text{Total occurrences} &= 10 \text{ (ones place)} + 10 \text{ (tens place)} \\ &= 20 \end{aligned}

Answer

(i) 20

(ii) Among the numbers 1–1000, how many times will the digit ‘7’ occur?

Step 1 · Count occurrences of '7' across place values from 1 to 1000

1. In the ones place: The digit $7$ appears $10$ times in every group of $100$ . Since there are $10$ hundreds from $1$ to $1000$ :

\begin{aligned} &10 \text{ times per hundred} \times 10 \text{ hundreds} \\ &= 100 \end{aligned}

2. In the tens place: The digit $7$ appears $10$ times in every group of $100$ ( $70\text{--}79, 170\text{--}179, \dots$ ). For $10$ hundreds:

\begin{aligned} &10 \text{ times per hundred} \times 10 \text{ hundreds} \\ &= 100 \end{aligned}

3. In the hundreds place: The digit $7$ appears in all numbers from $700$ to $799$ :

\begin{aligned} &799 - 700 + 1 \\ &= 100 \end{aligned}

Total occurrences from 1 to 1000:

\begin{aligned} \text{Total} &= 100 \text{ (ones)} + 100 \text{ (tens)} + 100 \text{ (hundreds)} \\ &= 300 \end{aligned}

Answer

(ii) 300

Common Mistakes

Numbers vs. Occurrences: Counting how many numbers contain the digit $7$ ( $19$ numbers in $1\text{--}100$ ) instead of counting every occurrence of the digit ( $20$ times in $1\text{--}100$ , because $77$ contains $7$ twice).
Missing Range Boundaries: When counting the numbers from $700$ to $799$ , forgetting to add $1$ (i.e. computing $799 - 700 = 99$ instead of $799 - 700 + 1 = 100$ ).

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