Line up the points, not the ends
To add 2.5 + 1.75 in columns, write the numbers one above the other so that the decimal points line up. Then ones sit under ones, tenths under tenths and hundredths under hundredths.
Lining up the right-hand ends instead would put the 5 tenths of 2.5 over the 5 hundredths of 1.75, and add pieces of two different sizes.
The points are lined up. 2.5 has nothing in the hundredths column, above the 5 of 1.75.
1.75 with its point out of line counts as 17.5: 2.5 + 17.5 = 20.0, not 2.5 + 1.75
Line up the decimal points
Lined up at the ends, the points sit a column apart and the columns read 1.75 as 17.5, so they add to 20.0. With the points in one column, every digit is in its own place and the sum is 4.25.
Fill the gap with a zero
2.5 stops at the tenths, so its hundredths place is empty. 2.5 has no hundredths, so write a 0 there. 2.50 is the same number as 2.5, because 5 tenths is 50 hundredths. Now every column has a digit in both rows.
The 0 fills the hundredths place of 2.50.
Add each column as usual
Start with the column on the right. The hundredths: 0 + 5 = 5. The tenths: 5 + 7 = 12 tenths, which is 1 one and 2 tenths. Write the 2 in the tenths column and carry the 1 to the ones column, just as 10 ones are carried as 1 ten.
The ones: 2 + 1 and the carried 1 make 4. Bring the point straight down into the answer, under the other points: 2.5 + 1.75 = 4.25.
An estimate checks it: 2.5 + 1.75 is a little less than 2.5 + 2 = 4.5, and 4.25 is a little less than 4.5.
The tenths made 12, so 2 is written and 1 is carried into the ones. The point comes straight down: 4.25.
Two slips
Lining up the ends turns 2.5 into 0.25, a tenth of itself, and gives 0.25 + 1.75 = 2.00. The points must line up.
Leaving out the carried 1 gives 3.25. The 12 tenths made one whole, and it belongs in the ones column.
Worked example: Spending in Stages with the Final Amount Known
Question Ali spent $3.75 on lunch. He then spent half of his remaining money on a book and had $6.40 left. (a) How much money did Ali have just before he bought the book? (b) How much money did Ali have at first?
1.Work backwards from the $6.40 he had left. The book cost half of his money, so the $6.40 is the other half.
The book cost half of what he had, so the $6.40 left is the other half. 2.(a) Just before he bought the book he had 2 × 6.40 = $12.80.
(a) Before the book he had 2 × 6.40 = $12.80. 3.Before that he spent $3.75 on lunch, so add it back: 12.80 + 3.75 = $16.55.
(b) Before lunch he had 12.80 + 3.75 = $16.55. 4.(b) Ali had $16.55 at first. Check: 16.55 − 3.75 = 12.80, and half of 12.80 is 6.40.
Answer: (a) $12.80; (b) $16.55
Common mistakes
- Undoing the steps in the order they happened: adding $3.75 to $6.40 first and then doubling gives $20.30. The last step, the book, must be undone first.
- Halving $6.40 instead of doubling it. He spent half, so the $6.40 he kept is the other half of what he had.