A jar has 5-cent coins and 10-cent coins. There are 9 more 5-cent… Easy

The question

Question: A jar has 5-cent coins and 10-cent coins.

There are 9 more 5-cent coins than 10-cent coins.

The coins are worth $2.25.

How many of each coin are in the jar?

1 Find what we know

What do we know?

Two kinds of coins: 5 cents and 10 cents each.

There are 9 more 5-cent coins than 10-cent coins.

All the coins are worth $2.25.

What do we want?

How many 10-cent coins, and how many 5-cent coins.

2 Name the unknown

We know the 5-cent coins from the 10-cent coins.

So we start with the 10-cent coins.

We call how many there are x.

9 more: x + 9.

We work in cents, so every number is whole.

$1 is 100 cents.

So $2.25 is 225 cents.

Careful!

There are 9 more 5-cent coins than 10-cent coins.

So the + 9 goes with the 5-cent coins, not the 10-cent coins.

Let's put it in a table: how many coins, the value of one, and the money in cents.

10-cent coins: x of them, 10 cents each.

5-cent coins: x + 9 of them, 5 cents each.

Money = how many times the value of one coin.

10 times x is 10x.

5 times all of x + 9 is 5(x + 9).

We keep x + 9 together in brackets.

The total: 225 cents in all.

3 Write the equation

In words: the 10-cent money plus the 5-cent money makes all the money.

So the equation is 10x + 5(x + 9) = 225.

Now we solve it.

4 Open the brackets

The 5 in front of the bracket multiplies every part inside.

So we write 5 times each part.

Work out each multiplication.

5 Group like terms

Terms are the parts between the + and − signs.

Like terms are the same kind: same letters, same powers.

All the x terms go together.

10x + 5x = 15x.

Now we have 15x + 45 = 225.

We want x by itself.

Two things are stuck to x: times 15 and + 45.

The + 45 is on the outside, so it comes off first.

Then the times 15.

6 Remove the + 45

To remove + 45, we take away 45.

We do it on both sides, so they stay equal.

+ 45 and − 45 make zero.

Cross them out.

225 − 45 = 180.

Now we have 15x = 180.

7 Remove the 15

15x means 15 times x.

We want just x.

To remove times 15, we divide by 15.

We do it on both sides, so they stay equal.

15 divided by 15 is 1.

So just x is left.

180 divided by 15 is 12.

So x = 12!

8 Answer the question

5-cent coins: x + 9.

12 + 9 = 21.

So 12 coins of 10 cents, and 21 coins of 5 cents.

9 Check

Let's check in the story.

21 is 9 more than 12, like the story says.

The money: 120 cents plus 105 cents makes 225 cents.

225 cents is $2.25, like the story says.

Does it make sense?

Both are whole numbers, not below 0.

That fits coins.

So the answer is right!

Let's look back.

First: find what we know.

Then: name the unknown.

Then: write the equation.

Then: solve it.

Then: answer the question.

Last: check.

▶ Watch it solved

A jar has 10-cent coins and 25-cent coins. There are twice as many… Medium

The question

Question: A jar has 10-cent coins and 25-cent coins.

There are twice as many 10-cent coins as 25-cent coins.

The coins are worth $3.60.

How many of each coin are in the jar?

1 Find what we know

What do we know?

Two kinds of coins: 10 cents and 25 cents each.

There are 2 times as many 10-cent coins as 25-cent coins.

All the coins are worth $3.60.

What do we want?

How many 25-cent coins, and how many 10-cent coins.

2 Name the unknown

We know the 10-cent coins from the 25-cent coins.

So we start with the 25-cent coins.

We call how many there are x.

There are 2 times as many 10-cent coins as 25-cent coins.

2 times x: 2x.

We work in cents, so every number is whole.

$1 is 100 cents.

So $3.60 is 360 cents.

Careful!

There are 2 times as many 10-cent coins as 25-cent coins.

So the 10-cent coins are 2x, not the 25-cent coins.

Let's put it in a table: how many coins, the value of one, and the money in cents.

25-cent coins: x of them, 25 cents each.

10-cent coins: 2x of them, 10 cents each.

Money = how many times the value of one coin.

25 times x is 25x.

10 times all of 2x is 10(2x).

We keep 2x together in brackets.

The total: 360 cents in all.

3 Write the equation

In words: the 25-cent money plus the 10-cent money makes all the money.

So the equation is 25x + 10(2x) = 360.

Now we solve it.

4 Open the brackets

The 10 in front of the bracket multiplies every part inside.

So we write 10 times each part.

Work out each multiplication.

5 Group like terms

Terms are the parts between the + and − signs.

Like terms are the same kind: same letters, same powers.

All the x terms go together.

25x + 20x = 45x.

Now we have 45x = 360.

6 Remove the 45

45x means 45 times x.

We want just x.

To remove times 45, we divide by 45.

We do it on both sides, so they stay equal.

45 divided by 45 is 1.

So just x is left.

360 divided by 45 is 8.

So x = 8!

7 Answer the question

10-cent coins: 2x.

2 × 8 = 16.

So 8 coins of 25 cents, and 16 coins of 10 cents.

8 Check

Let's check in the story.

2 times 8 is 16: 2 times as many, like the story says.

The money: 200 cents plus 160 cents makes 360 cents.

360 cents is $3.60, like the story says.

Does it make sense?

Both are whole numbers, not below 0.

That fits coins.

So the answer is right!

Let's look back.

First: find what we know.

Then: name the unknown.

Then: write the equation.

Then: solve it.

Then: answer the question.

Last: check.

▶ Watch it solved

A jar has 5-cent coins and 25-cent coins. There are 4 times as many… Harder

The question

Question: A jar has 5-cent coins and 25-cent coins.

There are 4 times as many 5-cent coins as 25-cent coins.

The coins are worth $5.40.

How many of each coin are in the jar?

1 Find what we know

What do we know?

Two kinds of coins: 5 cents and 25 cents each.

There are 4 times as many 5-cent coins as 25-cent coins.

All the coins are worth $5.40.

What do we want?

How many 25-cent coins, and how many 5-cent coins.

2 Name the unknown

We know the 5-cent coins from the 25-cent coins.

So we start with the 25-cent coins.

We call how many there are x.

There are 4 times as many 5-cent coins as 25-cent coins.

4 times x: 4x.

We work in cents, so every number is whole.

$1 is 100 cents.

So $5.40 is 540 cents.

Careful!

There are 4 times as many 5-cent coins as 25-cent coins.

So the 5-cent coins are 4x, not the 25-cent coins.

Let's put it in a table: how many coins, the value of one, and the money in cents.

25-cent coins: x of them, 25 cents each.

5-cent coins: 4x of them, 5 cents each.

Money = how many times the value of one coin.

25 times x is 25x.

5 times all of 4x is 5(4x).

We keep 4x together in brackets.

The total: 540 cents in all.

3 Write the equation

In words: the 25-cent money plus the 5-cent money makes all the money.

So the equation is 25x + 5(4x) = 540.

Now we solve it.

4 Open the brackets

The 5 in front of the bracket multiplies every part inside.

So we write 5 times each part.

Work out each multiplication.

5 Group like terms

Terms are the parts between the + and − signs.

Like terms are the same kind: same letters, same powers.

All the x terms go together.

25x + 20x = 45x.

Now we have 45x = 540.

6 Remove the 45

45x means 45 times x.

We want just x.

To remove times 45, we divide by 45.

We do it on both sides, so they stay equal.

45 divided by 45 is 1.

So just x is left.

540 divided by 45 is 12.

So x = 12!

7 Answer the question

5-cent coins: 4x.

4 × 12 = 48.

So 12 coins of 25 cents, and 48 coins of 5 cents.

8 Check

Let's check in the story.

4 times 12 is 48: 4 times as many, like the story says.

The money: 300 cents plus 240 cents makes 540 cents.

540 cents is $5.40, like the story says.

Does it make sense?

Both are whole numbers, not below 0.

That fits coins.

So the answer is right!

Let's look back.

First: find what we know.

Then: name the unknown.

Then: write the equation.

Then: solve it.

Then: answer the question.

Last: check.

▶ Watch it solved