Counting, Tens & Sharing — Free Math Kangaroo Lesson (Grades 1–4)

About this topic

A number is a way to say how many. How many cookies. How many balloons. How many feet on a wiggly bug!

In here you will count and never miss one. You will tie things into tidy tens. You will add, take away, and share into fair groups.

Go slow. Use your fingers. Draw little dots. That is real math, and it wins.

CHAPTER 1

Count and don't miss any

THEORY

Count these frogs out loud. Did you get 7? Count again. Did you get 7 both times?

The danger is counting one frog two times, or skipping one. So here is the safe way. Touch each frog once. Say the number. Then cross it off in your head before you move on.

When there are a lot of things, make little groups of 5. Five is easy to see in one look. Then skip-count: 5, 10, 15…

Two full groups of 5 is 10. Then 3 more makes 13.

🎯 Try it

You have two groups of 5 dots and 2 more dots. How many dots in all?

Here's how: 5 and 5 makes 10. Then 2 more. 10, 11, 12. The answer is 12.

Count both ends — the fence-post trick

Here is a sneaky one. Look at a little fence. Count the posts. Then count the gaps between them.

4 posts, but only 3 gaps. There is always one fewer gap than posts.

So when you count the houses from one number to another, you must count both ends. How many houses from 5 to 8? Touch them: 5, 6, 7, 8. That is 4 houses, not 3!

It is 8 − 5 + 1. The take-away gives the gaps. The + 1 puts back the first house.

🎯 Try it

How many numbers are there from 10 to 15? (Count both ends.)

Here's how: Touch them: 10, 11, 12, 13, 14, 15. That is 6 numbers. Or 15 − 10 = 5 gaps, then add 1 for the first one. 6.

A real Kangaroo problem (2016 Felix #5): Johannes gives a flyer to houses 15 to 47. How many houses?

Peek at the answer

Both end houses get a flyer, so count both ends. 47 − 15 = 32 gaps. Add 1 for the first house: 32 + 1 = 33 houses. Forget the + 1 and you get 32 by mistake — that is the trap!

THE TRICK

Touch each thing once and cross it off in your head. For big piles, make groups of 5 and skip-count. To count a row, the gaps are one fewer than the things.

WATCH OUT

When things are close together it is easy to count one of them twice. Cross each one off as you touch it so you never count it again. And when you count from one number to another, remember to count both ends.

WORKED EXAMPLE

PROBLEM · 2015 #4

How many dots do all ladybirds have together?

A) 17 B) 18 C) 19 D) 20 E) 21

You want all the dots on all the ladybirds. Do not try to see them all at once. Take one ladybird at a time.

Touch the first ladybird. Count its dots, say the number, keep it in your head. Now add the next ladybird's dots on. Keep going, bug by bug.

Go slow and add them one bug at a time. The total comes to 19 dots (choice C).

I point at the first ladybird and hold its number in my head. Then I add the next ladybird's dots on, and the next. Slow and steady wins here — if I rush, I might count a dot twice.

Answer: C — 19

MORE LIKE THIS

2016 · #2 In a cave there live a starfish, two seahorses and three turtles. They are visited by three starfish, four turtles and five seahorses....

In a cave there live a starfish, two seahorses and three turtles. They are visited by three starfish, four turtles and five seahorses. How many animals are there now in the cave altogether?

Show answer

Answer: E — 18

Show hints

Hint 1 of 2

Add up everyone who is now inside the cave: the original animals plus the visitors.

Still stuck? Show hint 2 →

Hint 2 of 2

Total the residents (1 + 2 + 3) and the visitors (3 + 4 + 5) separately, then combine.

Show solution

Approach: add residents and visitors

Living there: 1 starfish + 2 seahorses + 3 turtles = 6 animals.
Visitors: 3 starfish + 4 turtles + 5 seahorses = 12 animals.
Altogether 6 + 12 = 18 animals.

2016 · #4 Ten friends go to Robert’s birthday party. Six of them are girls. How many boys in total are at the party?

Ten friends go to Robert’s birthday party. Six of them are girls. How many boys in total are at the party?

Show answer

Answer: B — 5

Show hints

Hint 1 of 2

Don't forget whose party it is — the birthday child is there too.

Still stuck? Show hint 2 →

Hint 2 of 2

Count the boys among the ten friends, then add Robert himself.

Show solution

Approach: subtract girls, then add the host

Of the ten friends, 6 are girls, so 10 − 6 = 4 are boys.
Robert, the birthday boy, is also there, so add 1.
Total boys at the party: 4 + 1 = 5.

2016 · #5 Johannes hands out flyers to the houses with the numbers 15 to 47. How many houses get a flyer?

Johannes hands out flyers to the houses with the numbers 15 to 47. How many houses get a flyer?

Show answer

Answer: C — 33

Show hints

Hint 1 of 2

Counting from 15 to 47 includes both end houses.

Still stuck? Show hint 2 →

Hint 2 of 2

The count is the difference of the numbers plus one.

Show solution

Approach: count an inclusive range

The houses are numbered 15, 16, ..., 47.
Count = 47 − 15 + 1 = 33 houses get a flyer.

CHAPTER 2

Tens and ones

THEORY

Quick — how many sticks is this? Hard to tell fast, right?

Ten is your best friend here. Tie ten sticks into one bundle. Now the same pile is easy.

One bundle and 3 loose sticks is 13. The bundle is the tens. The loose sticks are the ones. Now you can count fast: 10… 11, 12, 13.

Look at the number 24. The 2 means 2 bundles of ten. The 4 means 4 loose ones.

So 24 is the same as 20 + 4. The tens come first, then the ones.

🎯 Try it

You have 3 bundles of ten and 5 loose ones. What number is that?

Here's how: Count the bundles by tens: 10, 20, 30. Then 5 loose ones. That is 35.

Bundles make a bigger number

Why do we love bundles? Because you can grab more by filling the tens spot. Two two-digit numbers, like 53 and 35, use the same digits — but the one with the bigger tens wins.

So to build the biggest number, put your biggest digit in the tens spot.

🎯 Try it

Use the digits 6 and 8 to make the biggest two-digit number you can. What is it?

Here's how: The tens spot is worth more, so put the bigger digit (8) there. 8 tens and 6 ones is 86. (Not 68!)

Peek (for big kids)

Three-digit numbers work the same way. 124 is 1 bundle of a hundred, 2 bundles of ten, and 4 ones. Each spot to the left is ten times bigger than the one on its right.

THE TRICK

The left digit counts bundles of ten. The right digit counts loose ones. So 47 is 4 tens and 7 ones. To make a number big, give the tens spot your biggest digit.

WORKED EXAMPLE

PROBLEM · 2014 #14

Each of the digits 2, 3, 4 and 5 will be placed in a square. Then there will be two numbers, which will be added together. What is the biggest number that they could make?

Figure for Math Kangaroo 2014 Problem 14

A) 68 B) 77 C) 86 D) 95 E) 97

You get the digits 2, 3, 4 and 5. You build two 2-digit numbers and add them. You want the biggest sum.

Tens are worth way more than ones. So put your two biggest digits, 5 and 4, in the two tens spots. Put 3 and 2 in the ones spots.

That makes 53 and 42, or 52 and 43 — either way the tens add to 5 + 4 = 9 tens, which is 90. Add the ones: 3 + 2 = 5. So 90 + 5 = 95 (choice D).

I do not try every mix. I just ask: where does a digit count most? In the tens spot. So my two biggest digits go up top, and the sum has to be 95.

Answer: D — 95

MORE LIKE THIS

2025 · #1 Nico and his little sister play with shells and marbles. Each shell is worth 6 and each marble is worth 1 (shell = 6, marble = 1). Which...

Nico and his little sister play with shells and marbles. Each shell is worth 6 and each marble is worth 1 (shell = 6, marble = 1). Which picture shows the value 16?

Show answer

Answer: E

Show hints

Hint 1 of 2

Each shell counts as 6 and each marble counts as 1 - add up each picture's total.

Still stuck? Show hint 2 →

Hint 2 of 2

You need a total of exactly 16, so look for two shells plus four marbles.

Show solution

Approach: add the shell and marble values in each option

A shell is worth 6 and a marble is worth 1.
Two shells give 12, and you need 4 more to reach 16, so 4 marbles.
The picture with two shells and four marbles totals 12 + 4 = 16.
That is option E.

2019 · #2 The diagram shows the number 8. A dot stands for the number 1 and a line for the number 5. Which diagram represents the number 12?

The diagram shows the number 8. A dot stands for the number 1 and a line for the number 5. Which diagram represents the number 12?

Show answer

Answer: C

Show hints

Hint 1 of 2

A dot is worth 1 and a line is worth 5; the example 8 uses 3 dots and 1 line.

Still stuck? Show hint 2 →

Hint 2 of 2

Make 12 using as many 5s (lines) as possible, then add dots for the rest.

Show solution

Approach: build 12 from fives and ones

A line = 5 and a dot = 1.
Two lines give 10, and two dots give 2, for a total of 12.
The diagram with 2 lines and 2 dots is C.

2025 · #4 Simona writes the numbers 2, 0, 2 and 5 in the boxes, one number per box (see picture). In what order can she write them so that the...

Simona writes the numbers 2, 0, 2 and 5 in the boxes, one number per box (see picture). In what order can she write them so that the calculation gives the biggest result?

Show answer

Answer: E — 5, 2, 0, 2

Show hints

Hint 1 of 2

The third box is the one that gets subtracted, so put the smallest number there.

Still stuck? Show hint 2 →

Hint 2 of 2

Put the 0 in the subtracted (third) box and add everything else.

Show solution

Approach: minimise what is subtracted, maximise what is added

The calculation is first + second minus third + fourth.
To make it biggest, subtract the smallest number, which is 0.
Then the other three (5, 2, 2) are all added: 5 + 2 - 0 + 2 = 9.
The order 5, 2, 0, 2 does this, which is option E.

★ MINI-QUIZ

Quick check: count and bundle

Two little problems about counting carefully and tens-and-ones.

2016 · #4 Ten friends go to Robert’s birthday party. Six of them are girls. How many boys in total are at the party?

Ten friends go to Robert’s birthday party. Six of them are girls. How many boys in total are at the party?

Show answer

Answer: B — 5

Show hints

Hint 1 of 2

Don't forget whose party it is — the birthday child is there too.

Still stuck? Show hint 2 →

Hint 2 of 2

Count the boys among the ten friends, then add Robert himself.

Show solution

Approach: subtract girls, then add the host

Of the ten friends, 6 are girls, so 10 − 6 = 4 are boys.
Robert, the birthday boy, is also there, so add 1.
Total boys at the party: 4 + 1 = 5.

2018 · #2 Susanne is 6 years old. Her sister Lisa is 2 years younger. Her brother Max is 2 years older than Susanne. How old are the three...

Susanne is 6 years old. Her sister Lisa is 2 years younger. Her brother Max is 2 years older than Susanne. How old are the three siblings altogether?

Show answer

Answer: D — 18

Show hints

Hint 1 of 2

Work out each sibling's age from Susanne's.

Still stuck? Show hint 2 →

Hint 2 of 2

Lisa is younger, Max is older — then just add the three ages.

Show solution

Approach: find each age, then add

Susanne is 6.
Lisa is 2 younger: 6 − 2 = 4.
Max is 2 older: 6 + 2 = 8.
Total: 6 + 4 + 8 = 18.

CHAPTER 3

Add and take away

THEORY

Watch this. 9 + 1 = 10. 8 + 2 = 10. 7 + 3 = 10. 6 + 4 = 10. See the pattern? Every pair that fills up to ten is a friend.

Tens are friendly. So when you add, fill one number up to ten first. Look at 8 + 5.

Take 2 from the 5 to fill the 8 up to a 10. Now you have 10 and 3 left. 10 + 3 = 13!

Hop on a number line

Another way: count on. Start at the big number and hop forward. To take away, hop back.

For 8 + 5: start at 8, then hop 9, 10, 11, 12, 13. Put up a finger each hop so you stop at the right place. To take away, like 13 − 4, hop back: 12, 11, 10, 9. The answer is 9.

🎯 Try it

Fill up to ten to add: 9 + 5 = ?

Here's how: Take 1 from the 5 to fill the 9 up to 10. Now 10 and 4 left. 10 + 4 = 14.

A real Kangaroo problem (2019 Felix #3): mother and Max weigh 60 kg together. Mother is 52 kg. How heavy is Max?

Peek at the answer

Max is the part that is left when you take the mother out of the 60. 60 − 52. Hop back, or fill up: 52 + 8 = 60. So Max is 8 kg.

THE TRICK

Fill up to ten first, then add the rest. Or hop along a number line: forward to add, back to take away.

WATCH OUT

Read the words! “Gives away” and “younger” mean take away. “More” and “altogether” mean add. Picking the wrong one is the most common slip.

WORKED EXAMPLE

PROBLEM · 2025 #1

Pablo has six balloons. He gives away two of his balloons. How many balloons does Pablo have now?

A) 2 B) 3 C) 4 D) 5 E) 6

Pablo has 6 balloons. He gives 2 away. “Gives away” means take away.

Hop back from 6, once for each balloon: 6, then 5, then 4. So 6 − 2 = 4 balloons (choice C).

I start at 6 and step back twice, once per balloon he gives away. 5, then 4. He has 4 left.

Answer: C — 4

MORE LIKE THIS

2018 · #2 Susanne is 6 years old. Her sister Lisa is 2 years younger. Her brother Max is 2 years older than Susanne. How old are the three...

Susanne is 6 years old. Her sister Lisa is 2 years younger. Her brother Max is 2 years older than Susanne. How old are the three siblings altogether?

Show answer

Answer: D — 18

Show hints

Hint 1 of 2

Work out each sibling's age from Susanne's.

Still stuck? Show hint 2 →

Hint 2 of 2

Lisa is younger, Max is older — then just add the three ages.

Show solution

Approach: find each age, then add

Susanne is 6.
Lisa is 2 younger: 6 − 2 = 4.
Max is 2 older: 6 + 2 = 8.
Total: 6 + 4 + 8 = 18.

2016 · #2 In a cave there live a starfish, two seahorses and three turtles. They are visited by three starfish, four turtles and five seahorses....

In a cave there live a starfish, two seahorses and three turtles. They are visited by three starfish, four turtles and five seahorses. How many animals are there now in the cave altogether?

Show answer

Answer: E — 18

Show hints

Hint 1 of 2

Add up everyone who is now inside the cave: the original animals plus the visitors.

Still stuck? Show hint 2 →

Hint 2 of 2

Total the residents (1 + 2 + 3) and the visitors (3 + 4 + 5) separately, then combine.

Show solution

Approach: add residents and visitors

Living there: 1 starfish + 2 seahorses + 3 turtles = 6 animals.
Visitors: 3 starfish + 4 turtles + 5 seahorses = 12 animals.
Altogether 6 + 12 = 18 animals.

2016 · #10 Together, Paul and Josef are 12 years old. How old will they both be together in four years’ time?

Together, Paul and Josef are 12 years old. How old will they both be together in four years’ time?

Show answer

Answer: E — 20

Show hints

Hint 1 of 2

You don't need each boy's age — only their combined age.

Still stuck? Show hint 2 →

Hint 2 of 2

In four years, both ages grow, so the total grows by 4 twice.

Show solution

Approach: track the combined age

Together they are 12 now.
In four years each is 4 years older, so their combined age rises by 4 + 4 = 8.
Together they will be 12 + 8 = 20.

CHAPTER 4

How many more, how many fewer

THEORY

Ann has 5 marbles. Ben has 3. Who has more? By how many? Don't add — just line them up and see who is left over.

Ben matches 3 of Ann's. Ann has 2 left with no partner. So Ann has 2 more.

The left-over pile is the same as a take-away. How many more = bigger minus smaller. Here, 5 − 3 = 2.

“How many fewer” is the very same gap, looked at from the other side. Ben has 2 fewer than Ann. Same 2.

🎯 Try it

Lily has 9 stickers. Sam has 6 stickers. How many more does Lily have?

Here's how: Match 6 of Lily's to Sam's 6. Lily has 3 left with no match. 9 − 6 = 3.

Fill the box so the total is right

Sometimes a puzzle hides a number. It tells you the boxes must add up to a certain total. You find the hidden one.

Here is the trick. Add the numbers you can see. Then ask: how much more do I need to reach the total? That gap is the hidden number.

You can see 4 and 3, that is 7. To reach 10 you need 3 more. So ? = 3.

A real Kangaroo problem (2023 Ecolier #2): the markers add up like 20 + 10 + 10 + ? + ? + 1 = 51, and the two question marks are the same number.

Peek at the answer

Add the ones you can read: 20 + 10 + 10 + 1 = 41. To reach 51 you still need 51 − 41 = 10. But that 10 is shared by two equal boxes. Split it fairly: 10 ÷ 2 = 5 in each.

🎯 Try it

Fill the box: 8 + ? = 14. What number is hidden?

Here's how: You have 8 and want 14. Count up: 9, 10, 11, 12, 13, 14. That is 6 hops. Or 14 − 8 = 6. The hidden number is 6.

Peek (for big kids): the giving trick

If one kid gives a marble to another, watch carefully: the giver loses one AND the taker gains one. So the gap between them shrinks by 2 for each marble given, not 1. To make two piles equal, give away half the gap. (That is why in 2015 Felix #11, a gap of 8 needs 4 sweets given, not 8.)

THE TRICK

How many more = bigger minus smaller. Line the two piles up and the part with no partner is the answer.

WATCH OUT

Watch for an extra person or extra step the words sneak in. In a “how many boys” party problem, the birthday child counts too! Read to the end before you answer.

WORKED EXAMPLE

PROBLEM · 2013 #15

At the London 2012 Olympic Games the USA won the most medals: 46 gold, 29 silver and 29 bronze. China was second with 38 gold, 27 silver and 23 bronze. How many more medals did the USA win than China?

A) 6 B) 14 C) 16 D) 24 E) 26

The USA won 46 + 29 + 29 medals. China won 38 + 27 + 23. We want how many more the USA got. Adding two big totals is heavy — so don't.

Match them up by color instead. Gold: 46 − 38 = 8 more. Silver: 29 − 27 = 2 more. Bronze: 29 − 23 = 6 more.

Now add just the little extras: 8 + 2 + 6 = 16 more medals (choice C).

Two big sums would be a lot of carrying. Instead I compare each color on its own — each is a tiny take-away — then add the three small gaps. Way lighter.

Answer: C — 16

MORE LIKE THIS

2021 · #11 Julie and Angela played “kangball”, a ball game. Each goal in their game scores 2 points. Julie scored 5 goals and Angela scored 9...

Julie and Angela played “kangball”, a ball game. Each goal in their game scores 2 points. Julie scored 5 goals and Angela scored 9 goals. How many more points than Julie did Angela score?

Show answer

Answer: C — 8

Show hints

Hint 1 of 3

First find how many MORE goals Angela scored than Julie.

Still stuck? Show hint 2 →

Hint 2 of 3

Each goal is worth 2 points, not 1.

Still stuck? Show hint 3 →

Hint 3 of 3

Turn just the extra goals into points.

Show solution

Approach: find the extra goals, then turn them into points

Angela scored 9 − 5 = 4 more goals than Julie.
Each goal is 2 points, so 4 goals is 4 × 2 = 8 points.
Angela scored 8 more points.

2015 · #11 Julia has 9 sweets and Katharina has 17 sweets. How many sweets does Katharina have to give to Julia so that they both have the same...

Julia has 9 sweets and Katharina has 17 sweets. How many sweets does Katharina have to give to Julia so that they both have the same amount of sweets?

Show answer

Answer: C — 4

Show hints

Hint 1 of 3

Katharina has more sweets than Julia, so she needs to give some away to make them even.

Still stuck? Show hint 2 →

Hint 2 of 3

Find the gap between their piles, then think about sharing that gap fairly.

Still stuck? Show hint 3 →

Hint 3 of 3

Whatever Katharina gives, Julia gains, so each sweet given closes the gap by two.

Show solution

Approach: share the difference equally

Line up the piles: Julia has 9 and Katharina has 17, so Katharina has 8 more.
Each time Katharina gives one sweet, she loses one and Julia gains one, so the gap shrinks by 2.
To close a gap of 8 the gap must shrink 4 times, so Katharina gives 4 sweets and both end with 13, choice C.

2022 · #15 The three zebras Runa, Zara and Biba take part in a competition. The winner is the zebra with the most stripes. Runa has 15 stripes....

The three zebras Runa, Zara and Biba take part in a competition. The winner is the zebra with the most stripes. Runa has 15 stripes. Zara has 3 stripes more than Runa. Runa has 5 stripes less than Biba. How many stripes does the winner have?

Show answer

Answer: C — 20

Show hints

Hint 1 of 2

Work out each zebra's stripe count from Runa's 15.

Still stuck? Show hint 2 →

Hint 2 of 2

The winner has the most stripes - compare all three.

Show solution

Approach: compute each total and take the largest

Runa has 15. Zara has 15 + 3 = 18. Biba has 15 + 5 = 20.
Biba has the most stripes.
So the winner has 20 stripes.

★ MINI-QUIZ

Quick check: add, take away, compare

Three short problems about putting together, taking away, and how many more.

2025 · #1 Pablo has six balloons. He gives away two of his balloons. How many balloons does Pablo have now?

Pablo has six balloons. He gives away two of his balloons. How many balloons does Pablo have now?

Show answer

Answer: C — 4

Show hints

Hint 1 of 2

You start with six and lose two of them.

Still stuck? Show hint 2 →

Hint 2 of 2

"Gives away" means take away — this is just subtraction.

Show solution

Approach: subtract what is given away

Pablo begins with 6 balloons.
He gives away 2, so take 2 away from 6.
6 − 2 = 4, so Pablo has 4 balloons.

2016 · #10 Together, Paul and Josef are 12 years old. How old will they both be together in four years’ time?

Together, Paul and Josef are 12 years old. How old will they both be together in four years’ time?

Show answer

Answer: E — 20

Show hints

Hint 1 of 2

You don't need each boy's age — only their combined age.

Still stuck? Show hint 2 →

Hint 2 of 2

In four years, both ages grow, so the total grows by 4 twice.

Show solution

Approach: track the combined age

Together they are 12 now.
In four years each is 4 years older, so their combined age rises by 4 + 4 = 8.
Together they will be 12 + 8 = 20.

2015 · #11 Julia has 9 sweets and Katharina has 17 sweets. How many sweets does Katharina have to give to Julia so that they both have the same...

Julia has 9 sweets and Katharina has 17 sweets. How many sweets does Katharina have to give to Julia so that they both have the same amount of sweets?

Show answer

Answer: C — 4

Show hints

Hint 1 of 3

Katharina has more sweets than Julia, so she needs to give some away to make them even.

Still stuck? Show hint 2 →

Hint 2 of 3

Find the gap between their piles, then think about sharing that gap fairly.

Still stuck? Show hint 3 →

Hint 3 of 3

Whatever Katharina gives, Julia gains, so each sweet given closes the gap by two.

Show solution

Approach: share the difference equally

Line up the piles: Julia has 9 and Katharina has 17, so Katharina has 8 more.
Each time Katharina gives one sweet, she loses one and Julia gains one, so the gap shrinks by 2.
To close a gap of 8 the gap must shrink 4 times, so Katharina gives 4 sweets and both end with 13, choice C.

CHAPTER 5

Equal groups and leftovers

THEORY

Look at these plates. 3 plates, 4 cookies on each. Count them all by fours: 4, 8, 12.

3 equal groups of 4 is 12. That is what 3 × 4 means — lots of equal groups.

Now the other way. You have 12 cookies and 3 friends. How many each? Deal them out like cards, one at a time, until they are gone. Each friend gets 4. That is sharing.

When it won't split fair

Things don't always share evenly. Say you have 13 cookies and 3 friends. Deal them: 4 each uses up 12. One cookie is left over.

We say 4 each, with 1 leftover. The leftover is the part that won't make another fair group.

🎯 Try it

Share 10 grapes fairly between 2 kids. How many grapes does each kid get?

Here's how: Deal them out: one for you, one for me, again and again. 10 grapes into 2 fair groups is 5 each.

Do the × before the +

What if a sum has both a + and a × in it? Like 2 × 3 + 4.

There is a rule everyone follows. Do the times first, then the plus. The × is a tight little group that sticks together.

First 2 × 3 = 6. Then 6 + 4 = 10. (Not 2 × 7!)

A real Kangaroo problem (2009 Ecolier #1): 2 × 9 + 200 + 9 = ?

Peek at the answer

Do the times first: 2 × 9 = 18. Now it is just adding: 18 + 200 + 9. Add the small ones first, 18 + 9 = 27, then 27 + 200 = 227.

🎯 Try it

Do the times first: 4 × 3 + 2 = ?

Here's how: First 4 × 3 = 12. Then 12 + 2 = 14. Do not add 3 + 2 first!

Peek (for big kids): add a staircase the fast way

To add 1 + 2 + 3 + 4 + 5 + 6, don't go left to right. Pair the ends: 1 + 6 = 7, 2 + 5 = 7, 3 + 4 = 7. Three sevens is 21! That is the lightest way to total a staircase of numbers.

THE TRICK

Times means equal groups. Sharing deals them out fairly. If some won't fit a fair group, that part is the leftover. And in a mixed sum, do the × before the +.

WATCH OUT

Don't forget the leftover. If you have 13 and make groups of 4, you get 3 groups and 1 left, not 4 groups.

WORKED EXAMPLE

PROBLEM · 2018 #3

Which beetle has to fly away so that the beetles that are left have 20 dots altogether?

A) Beetle with 4 dots B) Beetle with 7 dots C) Beetle with 5 dots D) Beetle with 6 dots E) no beetle

You want the beetles that stay to have 20 dots in all. One beetle has to fly away with the extra dots.

First count every beetle's dots and add them: 5, 7, 5, 6, 4. That is 5 + 7 + 5 + 6 + 4 = 27 dots in all.

You have 27 but want only 20 left. How many extra? 27 − 20 = 7. So the beetle that flies away must carry off exactly 7 dots.

Send away the beetle with 7 dots. The rest are 5 + 5 + 6 + 4 = 20. The answer is the beetle with 7 dots (choice B).

I add every beetle's dots first to get 27. Then I ask: how many over 20? That is 7. So I look for the beetle with exactly 7 dots and send it off.

Answer: B — Beetle with 7 dots

MORE LIKE THIS

2016 · #8 Grandma stands in the courtyard calling her cat and all her chickens. After a little while, 20 legs come running towards her. How many...

Grandma stands in the courtyard calling her cat and all her chickens. After a little while, 20 legs come running towards her. How many chickens does Grandma have?

Show answer

Answer: C — 8

Show hints

Hint 1 of 2

The cat has 4 legs; every chicken has 2.

Still stuck? Show hint 2 →

Hint 2 of 2

Take away the cat's legs first, then split the rest into pairs.

Show solution

Approach: remove the cat's legs, then divide by two

The cat accounts for 4 of the 20 legs, leaving 20 − 4 = 16 chicken legs.
Each chicken has 2 legs, so there are 16 ÷ 2 = 8 chickens.

2018 · #2 The same number of kangaroos should be in both parks. How many kangaroos have to be moved from the left park to the right park to make...

The same number of kangaroos should be in both parks. How many kangaroos have to be moved from the left park to the right park to make that happen?

Show answer

Answer: B — 5

Show hints

Hint 1 of 3

First count the kangaroos in each park and write the two numbers down.

Still stuck? Show hint 2 →

Hint 2 of 3

If you put all of them together and shared them fairly, how many would each park get?

Still stuck? Show hint 3 →

Hint 3 of 3

Move kangaroos one at a time from the crowded park until both parks match.

Show solution

Approach: share the kangaroos fairly between the two parks

Count: the left park has 13 kangaroos and the right park has 3, which is 16 kangaroos in all.
Sharing 16 fairly means 8 in each park.
The right park needs 5 more to reach 8, so move 5 kangaroos over — now both parks have 8.

2012 · #14 15 tables were set for a party. 5 plates were laid on 6 of the tables. 3 plates were laid on the rest of the tables. How many plates...

15 tables were set for a party. 5 plates were laid on 6 of the tables. 3 plates were laid on the rest of the tables. How many plates were needed in total?

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Answer: C — 57

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Hint 1 of 2

Find how many tables get 3 plates after 6 tables get 5 plates.

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Hint 2 of 2

Work out each group's plates and add them.

Show solution

Approach: split into two groups of tables

6 tables get 5 plates each: 6 x 5 = 30 plates.
The other 15 - 6 = 9 tables get 3 plates each: 9 x 3 = 27 plates.
Altogether 30 + 27 = 57 plates are needed.
The answer is 57.

⬢ FINAL TEST

Final Test

Five real Kangaroo problems, easy to medium. Take your time and count carefully!

2025 · #1 Pablo has six balloons. He gives away two of his balloons. How many balloons does Pablo have now?

Pablo has six balloons. He gives away two of his balloons. How many balloons does Pablo have now?

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Answer: C — 4

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Hint 1 of 2

You start with six and lose two of them.

Still stuck? Show hint 2 →

Hint 2 of 2

"Gives away" means take away — this is just subtraction.

Show solution

Approach: subtract what is given away

Pablo begins with 6 balloons.
He gives away 2, so take 2 away from 6.
6 − 2 = 4, so Pablo has 4 balloons.

2019 · #3 Mother kangaroo and her son Max together weigh 60 kg. The mother on her own weighs 52 kg. How heavy is Max?

Mother kangaroo and her son Max together weigh 60 kg. The mother on her own weighs 52 kg. How heavy is Max?

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Answer: B — 8 kg

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Hint 1 of 2

The two of them together make 60 kg, and the mother is part of that.

Still stuck? Show hint 2 →

Hint 2 of 2

Take the mother's weight away from the total to find Max.

Show solution

Approach: subtract the mother's weight from the combined weight

Together mother and Max weigh 60 kg.
The mother alone is 52 kg, so Max is what is left: 60 − 52.
60 − 52 = 8, so Max weighs 8 kg.
The answer is B.

2016 · #4 Ten friends go to Robert’s birthday party. Six of them are girls. How many boys in total are at the party?

Ten friends go to Robert’s birthday party. Six of them are girls. How many boys in total are at the party?

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Answer: B — 5

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Hint 1 of 2

Don't forget whose party it is — the birthday child is there too.

Still stuck? Show hint 2 →

Hint 2 of 2

Count the boys among the ten friends, then add Robert himself.

Show solution

Approach: subtract girls, then add the host

Of the ten friends, 6 are girls, so 10 − 6 = 4 are boys.
Robert, the birthday boy, is also there, so add 1.
Total boys at the party: 4 + 1 = 5.

2016 · #8 Grandma stands in the courtyard calling her cat and all her chickens. After a little while, 20 legs come running towards her. How many...

Grandma stands in the courtyard calling her cat and all her chickens. After a little while, 20 legs come running towards her. How many chickens does Grandma have?

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Answer: C — 8

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Hint 1 of 2

The cat has 4 legs; every chicken has 2.

Still stuck? Show hint 2 →

Hint 2 of 2

Take away the cat's legs first, then split the rest into pairs.

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Approach: remove the cat's legs, then divide by two

The cat accounts for 4 of the 20 legs, leaving 20 − 4 = 16 chicken legs.
Each chicken has 2 legs, so there are 16 ÷ 2 = 8 chickens.

2013 · #15 At the London 2012 Olympic Games the USA won the most medals: 46 gold, 29 silver and 29 bronze. China was second with 38 gold, 27 silver...

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Answer: C — 16

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Hint 1 of 3

Add up all of the USA's medals, then add up all of China's medals.

Still stuck? Show hint 2 →

Hint 2 of 3

The lighter way is to compare gold to gold, silver to silver, and bronze to bronze.

Still stuck? Show hint 3 →

Hint 3 of 3

Find each of those three differences and add the three differences together.

Show solution

Approach: compare each colour separately so the numbers stay small

Gold: the USA had 46 and China had 38, that is 8 more gold.
Silver: 29 against 27 is 2 more, and bronze: 29 against 23 is 6 more.
Adding the three extras, 8 + 2 + 6 = 16 more medals, which is answer C.

APPENDIX

Quick reference

Memorize these

Touch and cross off. Count both ends; the gaps are one fewer than the things.
10 = a full bundle. The left digit is tens, the right is ones. Make tens to add fast.
More / altogether means add. Gives away / fewer / younger means take away.
How many more = bigger − smaller.
Share by dealing out one at a time. The part that won't make a fair group is the leftover. Do × before +.

→ Practice Arithmetic & Operations problems