Rows and columns
Three children counted the books they read on Monday, Tuesday and Wednesday, and put the counts in a table. Each child has a row, running across, with the name at the start of it. Each day has a column, running down, with the day at the top of it. Every number in the table belongs to one row and one column.
How many books did Ben read on Tuesday? Start with the row: find Ben's name down the side.
Ben's row runs across the table and holds his three counts: 3, 6 and 1.
Where the row and the column cross
Now follow Ben's row along until you reach the Tuesday column. The number where the row and the column cross is 6, so Ben read 6 books on Tuesday.
Keep a finger on the row as you move along it. The 3 just before the 6 is Ben's Monday count, one column too early, and the 2 above the 6 is Ava's Tuesday count, one row too high. Both are next to the right cell, and both are wrong.
Ben's row and the Tuesday column cross at 6.
Totals
Add along a row for one person's total. Ben read 3 + 6 + 1 = 10 books across the three days.
Add down a column for one day's total. On Tuesday the three children read 2 + 6 + 4 = 12 books between them.
The last column adds along each row, and the last row adds down each column. Either way, all the books come to 30.
A missing number
A table with totals can have a gap filled in. Suppose Ben's Tuesday count is missing, but his total of 10 is given. His three days add up to 10, so the missing day is what is left after the other two are taken away: 10 − 3 − 1 = 6.
Check by adding: 3 + 6 + 1 = 10. The total itself is not the answer, because it counts all three days, and adding the two known days, 3 + 1 = 4, gives only the part that is not missing.
Ben's total is 10, and his Monday and Wednesday counts are 3 and 1, so the missing Tuesday count is 6.
Worked example: Two-Way Table with One Missing Number
Question A table shows the clubs that the children in one class chose. Of the boys, 12 chose Art and 15 chose Chess, which is 27 boys. Of the girls, 9 chose Chess, and there are 23 girls. The number of girls who chose Art is missing. In all, 26 children chose Art and 24 chose Chess, which is 50 children. (a) How many girls chose Art? (b) How many more girls than boys chose Art?
1.Find the row for the girls. It has the missing number, the 9 girls who chose Chess, and the total of 23 girls.
The missing number is in the row for the girls. That row ends with the total, 23. 2.The two clubs in a row add up to the row total, so the missing number is 23 − 9 = 14.
The two clubs add up to the row total, so the missing number is 23 − 9 = 14. 3.(a) 14 girls chose Art. Check with the Art column: 12 + 14 = 26, which is the total in the table.
(a) 14 girls chose Art. The Art column agrees: 12 + 14 = 26. 4.(b) Read down the Art column: 12 boys and 14 girls. 14 − 12 = 2, so 2 more girls than boys chose Art.
(b) In the Art column, 14 − 12 = 2 more girls than boys.
Answer: (a) 14 girls; (b) 2 more girls
Common mistakes
- Taking 9 from the total of the whole table, 50 − 9 = 41. The 50 counts every child in the class. The girls' row adds up to 23, so the subtraction starts from 23.
- Comparing the row totals for part (b), 27 − 23 = 4 more boys. Those totals count both clubs. The question is about Art only, so read the Art column.
Worked example: Price Table Where Buying More Costs Less Each
Question A shop sells notebooks, and the price of each notebook depends on how many are bought. For 1 to 5 notebooks, each costs $4. For 6 to 11 notebooks, each costs $3. For 12 or more notebooks, each costs $2. (a) Mrs Tan buys 7 notebooks. How much does she pay? (b) Mr Lee needs 11 notebooks. How much less does he pay if he buys 12 notebooks instead of 11?
1.Find the row for 7 notebooks. 7 is between 6 and 11, so each notebook costs $3.
7 notebooks are in the row for 6 to 11, so each costs $3. 2.(a) 7 × 3 = 21, so Mrs Tan pays $21.
(a) 7 × 3 = $21. 3.11 is in the same row, so 11 notebooks cost 11 × 3 = $33.
11 notebooks are in the same row: 11 × 3 = $33. 4.12 is in the row for 12 or more, so each notebook costs $2, and 12 notebooks cost 12 × 2 = $24.
12 notebooks are in the next row, at $2 each: 12 × 2 = $24. 5.(b) 33 − 24 = 9, so Mr Lee pays $9 less when he buys 12 notebooks.
(b) 33 − 24 = $9 less for 12 notebooks than for 11.
Answer: (a) $21; (b) $9 less
Common mistakes
- Using the first row for every order, 7 × 4 = $28. The price of $4 is only for 1 to 5 notebooks. Find the row that holds the number bought before multiplying.
- Pricing the first 5 notebooks at $4 and the other 2 at $3. The table gives one price for every notebook in the order, so all 7 notebooks cost $3 each.