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"And" multiplies, "or" adds

Counting and combinations · level 2, lesson 3 of 6

A question to start

The school café has 3 kinds of sandwich and 2 kinds of wrap. It also has 2 drinks.

  • Jada wants a sandwich or a wrap. How many choices does she have?
  • Omar wants a sandwich and a drink. How many choices does he have?

Both questions use the same café and almost the same words, but one answer is found by adding and the other by multiplying. One little word, "and" or "or", tells you which. By the end of this lesson you will know why, and you will be able to solve problems that need both.

The idea

"Or": one thing from one of the piles

Jada is choosing one thing. It can be from the sandwich pile or from the wrap pile. Imagine tipping both piles onto one big plate. Now she just picks one thing from the plate. How many things are on the plate? 3 sandwiches and 2 wraps: 3 + 2 = 5.

sandwich OR wrapone plate: 3 + 2 = 5 sandwich AND drink: 3 × 2 = 6 milkjuice

This is the addition rule: if you choose one thing, from one group or another group, and nothing is in both groups, add the sizes of the groups.

"And": one thing from each pile

Omar is choosing two things at once: one sandwich and one drink. His choice is a pair, just like Teddy's outfits in level 1. Every sandwich can go with every drink, so we get a box with 3 rows and 2 columns, or a tree with 3 branches each splitting into 2. Either way: 3 × 2 = 6.

This is the multiplication rule from the last lesson: if you choose one thing from the first group and then one thing from the second group, multiply.

Why the difference?

Here is the heart of it. Ask yourself: how many things does one answer contain?

  • Jada's answer is one thing (a cheese sandwich, say). The answers are just all the things there are, so we count the things: add.
  • Omar's answer is two things together (a cheese sandwich with milk). Each thing in the first group teams up with every thing in the second group, so the answers come in equal groups: multiply.

A good habit: before you count, say the choice as a sentence. "I pick a sandwich or a wrap": add. "I pick a sandwich and a drink": multiply. "I pick a sandwich and then a drink": multiply.

Using both

What about Kai, who wants (a sandwich or a wrap) and a drink? Do the "or" part first: there are 3 + 2 = 5 things to eat. Then the "and" part: each of those 5 goes with each of the 2 drinks. 5 × 2 = 10 meals.

Check it another way. Sandwich meals: 3 × 2 = 6. Wrap meals: 2 × 2 = 4. A meal is a sandwich meal or a wrap meal, so add: 6 + 4 = 10. ✓ Two ways, same answer.

Roads and routes

Here is a map. There are 3 roads from Ash to Birch, 2 roads from Birch to Cedar, and 1 road straight from Ash to Cedar through the woods.

AshBirchCedar through the woods

How many ways can you travel from Ash to Cedar, never going back?

  • Through Birch: a road to Birch and then a road to Cedar: 3 × 2 = 6 routes.
  • Through the woods: 1 route.
  • A route goes through Birch or through the woods: 6 + 1 = 7 routes.

A warning about "or": no double counting

The addition rule says "nothing is in both groups". Here is why that matters. In a toy box there are 4 red toys and 3 balls. How many toys are red or a ball? If one of the balls is red, it is in both lists. Adding 4 + 3 = 7 counts the red ball twice! The true answer is 7 − 1 = 6. When something could be in both groups, list carefully, or take away the ones you counted twice. (In later levels this idea grows into a big tool called inclusion-exclusion.)

Worked examples

1. Borrowing a book. The class library has 5 story books and 3 comics.

  • You may borrow one book, a story or a comic: 5 + 3 = 8 choices.
  • You may borrow one story and one comic: 5 × 3 = 15 choices.

2. Ice-cream again. You may have a cone with one scoop (chocolate, vanilla or mint) or a cup with two different scoops. How many treats?

Cones: 3. Cups with two different flavours: from level 1, there are 3 (chocolate and vanilla, chocolate and mint, vanilla and mint). Cone or cup: 3 + 3 = 6 treats.

3. Getting dressed for weather. On a sunny day you wear one of 4 T-shirts. On a rainy day you wear one of 2 raincoats. Either day, you add one of 3 hats. How many outfits are possible over both kinds of day?

Sunny outfits: 4 × 3 = 12. Rainy outfits: 2 × 3 = 6. Sunny or rainy: 12 + 6 = 18. (Or: (4 + 2) × 3 = 6 × 3 = 18. ✓)

4. Two-digit or one-digit? How many numbers from 1 to 99 have only odd digits?

One-digit numbers with an odd digit: 1, 3, 5, 7, 9, which is 5. Two-digit numbers: the tens digit is odd (5 ways) and the ones digit is odd (5 ways): 5 × 5 = 25. One-digit or two-digit: 5 + 25 = 30.

Try it

  1. A shop sells 6 kinds of apple and 4 kinds of pear. You buy one fruit. How many choices?
  2. In the same shop, you buy one apple and one pear. How many choices?
  3. For breakfast you have cereal (3 kinds) or toast (2 kinds), and also a drink (milk, juice or water). How many breakfasts?
  4. Use the map: a new road opens from Ash to Birch, so there are 4. How many routes from Ash to Cedar now?
  5. There are 5 girls and 4 boys in a group. The teacher picks one child to hand out paper. How many choices? Now she picks one girl and one boy to carry a box. How many choices?
  6. A bag has 3 green marbles, 5 big marbles, and nothing else. Two of the big marbles are green. How many marbles are in the bag?
  7. How many numbers from 10 to 99 have both digits even? (0 counts as even, but the tens digit cannot be 0.)
Answers
  1. One fruit, an apple or a pear: 6 + 4 = 10.
  2. An apple and a pear: 6 × 4 = 24.
  3. Food: 3 + 2 = 5. Then a drink: 5 × 3 = 15 breakfasts.
  4. Through Birch: 4 × 2 = 8. Through the woods: 1. Total 9 routes.
  5. One child: 5 + 4 = 9. One girl and one boy: 5 × 4 = 20.
  6. 3 + 5 = 8 counts the 2 big green marbles twice. 8 − 2 = 6 marbles. (Check: 2 big green, 1 small green, 3 big not-green: 6.)
  7. Tens digit: 2, 4, 6 or 8 (4 ways). Ones digit: 0, 2, 4, 6 or 8 (5 ways). 4 × 5 = 20.

Common mistake

Adding when you should multiply. "3 sandwiches and 2 drinks, so 5 lunches" is the same mistake as Teddy's "2 shirts and 3 pants, so 5 outfits". If one answer has two things in it, you need to multiply.

Adding groups that overlap. "Red or a ball" counts the red ball twice. Add only when no thing is in both groups, or take the doubles away.

Where it leads