Exercises — Place value system — units, tens, hundreds, thousands, lakhs, crores

Read right → left, ticking each column name off the ladder: $8_{\text{units}}{2}_{\text{tens}}{4}_{\text{hundreds}}{3}_{\text{thousands}}{7}_{\text{ten thousands}}{6}_{\text{lakhs}}$ What we did: matched each digit to a box on the ladder figure, starting from units. Answer: $7$ sits in the ten thousands column ($i=4$, worth $10^4$).

Group 3-2-2-2 from the right (parent-note comma rule): $3,00,70,005$.

• Top group $3$ → crores
• next two $00$ → lakhs = zero lakhs
• next two $70$ → thousands = seventy thousand
• last three $005$ → five Answer: "Three crore, seventy thousand, five"

Locate the column first (why: the rule needs $10^i$): $5_{\text{units}}{7}_{\text{tens}}{6}_{\text{hundreds}}{8}_{\text{thousands}}$ So $8$ is in thousands, $i=3$, worth $10^3=1000$. $\text{place value}=8\times 10^3=8\times 1000=8000$

Split digit by digit, each into (digit) $\times$ (its power): $50204=5\times 10^4+0\times 10^3+2\times 10^2+0\times 10^1+4\times 10^0$ $=50000+0+200+0+4=50204✓$ Why the two zeros stay: each $0$ is a placeholder — it holds its box open so the $5$ stays in ten-thousands and the $2$ stays in hundreds. Drop them and the number collapses. See Expanded form and standard form.

Read right → left: $1,4,3,2,5,6$ → the $6$ is the 6th column, i.e. $i=5$, the lakhs place. $6\times 10^5=6\times 1,00,000=6,00,000$

From Q3: place value of $8$ is $8000$; its face value is just $8$. $8000-8=7992$ Why they differ: the column ($10^3$) stretches the digit's worth. Face value is the digit "bare"; place value is the digit "wearing its column".

Same face value ($4$), different columns — that is the whole point of place value.

• Rightmost $4$: units, $4\times 10^0=4$.
• Leftmost $4$: it is the 6th column ($i=5$), lakhs, $4\times 10^5=4,00,000$. $400000-4=3,99,996$

Place value of the $0$: it sits in tens, so $0\times 10^1=0$ — it adds nothing to the total. But deleting it is a disaster: $504$ becomes $54$. The $5$ falls from hundreds ($5\times 100=500$) down to tens ($5\times 10=50$). The number drops by $450$ even though the $0$'s place value was $0$. Lesson: $0$ contributes $0$ to the sum but everything to the positions — it holds the column open.

To make a number large, put the biggest digit in the most valuable column (thousands, $10^3$) and shrink from there.

• Largest: sort digits big→small: $7,5,3,0\Rightarrow 7530$. To make it small, put the smallest digit in the biggest column — but a 4-digit number cannot start with $0$ (that would make it a 3-digit number $0753=753$). So put the smallest non-zero digit ($3$) in thousands, then the rest ascending:
• Smallest: $3,0,5,7\Rightarrow 3057$. See Comparing and Ordering Numbers for the ordering logic.

A 6-digit number has columns (right→left): units, tens, hundreds, thousands, ten-thousands, lakhs ($10^5$, the leftmost).

• Fixed: lakhs $=4$, units $=9$.
• To make the whole thing smallest, fill every free column with the smallest digit $0$. $400009\Rightarrow 4,00,009$ Check it's 6 digits: leftmost column is $4\ne 0$, good. Reads "four lakh nine".

A 7-digit number's columns (right→left, $i=0\dots 6$): units, tens, hundreds, thousands, ten-thousands ($10^4$), lakhs ($10^5$), ten-lakhs ($10^6$)… wait — that's only 7 columns and the crores place is $10^7$ (the 8th). So a number with a crores digit has 8 digits. Re-reading: the clue lists crores, ten-lakhs, lakhs, ten-thousands and units — filling an 8-digit ladder, everything else $0$: $30070005$ Grouping 3-2-2-2: $3,00,70,005$. Reading: "Three crore, seventy thousand, five" Place value of the 7: it sits in ten-thousands, $7\times 10^4=70000$.

Work backwards from each place value to its digit:

• thousands: $d\times 10^3=6000\Rightarrow d=6$.
• hundreds: $d\times 10^2=400\Rightarrow d=4$.
• tens and units are $0$ (no other non-zero digits). $N=6\times 10^3+4\times 10^2+0\times 10^1+0\times 10^0=6400$ Answer: $N=6400$, expanded form $6\times 10^3+4\times 10^2+0\times 10^1+0\times 10^0$.

• In $295000$: read right→left $0,0,0,5,9,2$ → $9$ is the 5th column ($i=4$), ten-thousands: $9\times 10^4=90000$.
• In $30900$: read right→left $0,0,9,0,3$ → $9$ is the 3rd column ($i=2$), hundreds: $9\times 10^2=900$. $\frac{90000}{900}=100$ Why the answer is a clean power of ten: both are the same face value $9$, so the ratio is just $10^4/10^2=10^2=100$ — the columns differ by exactly two steps on the ladder.

Strip the face value off first. $80,00,000=8\times 10,00,000=8\times 10^6$.

• face value $=8$.
• power of ten $=10^6$, i.e. $i=6$.
• the $10^6$ column is ten-lakhs (or "ten million" in the international system). Answer: column = ten-lakhs, power $=10^6$, face value $=8$.