Lesson Guide
Explanation
Inside a number, a digit is never just itself. It is a count of whatever column it stands in. In 30 060 the 3 stands in the ten-thousands column, so it is three ten-thousands: thirty thousand. The 6 stands in the tens column, so it is six tens: sixty. When a question compares two digits it is always comparing their values, and the first move is to write those values down. Three and six are two apart; thirty thousand and sixty are a world apart.
Your Turn
In 7 205, write the value of the 7 and the value of the 2. Check: 7000 and 200.
Watch Out
A zero in the middle changes every value to its left. In 30 060 the 3 is worth 30 000 only because of the two zeros after it.
Explanation
Moving one column to the left multiplies a digit's value by ten. Two columns, a hundred. Three columns, a thousand. So when the same digit appears twice in a number, the ratio between the two values is simply a power of ten: count the columns between them and that is how many zeros. The 7 in 7 000 sits two columns left of the 7 in 70, so it is worth 100 times as much. When the digits are the same, that is the whole method. When they are different, there is one more step, and that is next.
Your Turn
In 404 040, how many times larger is the value of the first 4 than the value of the last 4? Check: the first 4 is 400 000, the last is 40 — four columns apart, so 10 000 times.
Watch Out
Count the columns between the digits, not the digits between them. From the tens to the ten-thousands is three steps, not two.
Explanation
When the two digits are different, the columns alone do not settle it, because a 3 in one column and a 6 in another are not the same size to begin with. So you divide: how many times larger is thirty thousand than sixty? Thirty thousand divided by sixty. Drop a zero from each and it is three thousand divided by six, which is five hundred. Or build up from sixty: five sixties are three hundred, fifty sixties are three thousand, five hundred sixties are thirty thousand. Either way, the 3 is worth five hundred times the 6. 'How many times larger' is always a division of the two values.
Your Turn
In 40 800, how many times larger is the value of the 4 than the value of the 8? Check: 40 000 ÷ 800 = 400 ÷ 8 = 50.
Watch Out
Divide the big value by the small one, not the big digit by the small digit. 3 ÷ 6 is the wrong sum, and so is 6 ÷ 3.
Explanation
There is a faster route once you see how the two ideas combine. First the columns: from the tens to the ten-thousands is three steps, so the columns alone would make the 3 worth a thousand times the 6. Then the digits: a 3 is half of a 6, so halve it. A thousand halved is five hundred. This is exactly the same answer as the division, and it lets you handle a question like 'the 8 in 800 000 against the 2 in 2 000' in two thoughts: three columns is a thousand, and 8 is four times 2, so four thousand.
Your Turn
In 9 040 000, the 9 against the 4: how many times? Check: millions to ten-thousands is 2 steps → × 100; 9 is 2¼ times 4 → 225. (Or 9 000 000 ÷ 40 000 = 225.)
Watch Out
If the digit on the left is SMALLER than the digit on the right, the adjustment is a fraction: the 2 in 2 000 against the 8 in 80 is 100 × ¼ = 25. It is still "times larger", because the column wins.
Explanation
The same idea runs in reverse: a number is described by the ratio between two of its digits and you have to find it. 'The 7 is worth 100 times the other 7' means the two 7s are two columns apart, with exactly one column between them. Check each option by counting columns: 7 750 has its sevens one apart, ten times; 7 075 has them two apart, a hundred times, so that is the one; 70 007 has them four apart, ten thousand times. Column counting turns this from a puzzle into a checklist. Quiz and exercises now.
Your Turn
Which has a 5 worth 1000 times its other 5: 5 500, 5 050, 5 005 or 55 000? Check: 5 005 — thousands to ones is three columns.
Watch Out
"100 times" is two columns apart, not a hundred columns and not one. Ten has one zero, one column; a hundred has two.
📌 Key Points — this lesson in a nutshell
- A digit's value is the digit times its column: the 3 in 30 060 is worth 30 000, not 3.
- Each column is 10 times the one to its right. Three columns apart means 1000 times.
- "How many times larger" is a division: value ÷ value. 30 000 ÷ 60 = 500.
- Same digit, different columns: the ratio is a power of ten. Different digits: work the division out — 40 000 ÷ 800 = 50.
Step 1: Introduction
Step 1 of 5
Was this lesson helpful? Rate this Course
Discussion
Loading comments...