Chapter 7 of 8

All 8 chapters
  1. 1Letters for unknown numbers, readunderstood
  2. 2Putting numbers in, readunderstood
  3. 3Collecting like terms, readunderstood
  4. 4Brackets, readunderstood
  5. 5Solving equations in one step, readunderstood
  6. 6Solving equations in two steps, readunderstood
  7. 7Writing your own equations, readunderstood
  8. 8Sequences and the nth term, readunderstood

M3.8.7 Chapter 7

Writing your own equations

Three pieces of rope of the same length and a 5 m piece are joined. The whole rope is 26 m long. How long is each of the 3 pieces?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Write an equation from a word problem and solve it
  • Check an answer by putting it back into the equation

A word problem tells a story with numbers. To solve it, turn the story into an equation. Then solve the equation, and check the answer.

Words and symbols mean the same thing. This table shows how some words are written with a letter.

A table with two columns, words and symbols. The rows are: add 5 to n, n + 5. 3 times n, 3n. Take 7 from n, n − 7. Double n, add 6, 2n + 6. Is the same as, =.wordssymbolsadd 5 to nn + 53 times n3ntake 7 from nn − 7double n, add 62n + 6is the same as=
Take care with the order: "take 7 from n" is n − 7, not 7 − n.

Here are four steps. First, pick a letter for the unknown, and say what it means. Second, write the equation. Third, solve it. Fourth, check the answer.

Worked example

Joseph has some beans. He gives 7 away and has 12 left. How many beans did he have at the start?

  1. Let n be the beans Joseph had at the start.
  2. He gave away 7 and has 12 left, so n − 7 = 12.
  3. Add 7 to both sides: n = 12 + 7 = 19.
  4. Check: 19 − 7 = 12. It is true. He had 19 beans.

Answer: 19 beans

One long bar with the letter n over it. It is cut into two parts. The left part is 12 beans, the beans left. The right part is 7 beans, the beans given away.12 beans7 beansn
The whole bar is n. The 7 given away and the 12 left make n.

Draw it first

From a story to an equation

  1. Step 1: a bar with 3 equal shaded parts, each labelled x, and one part labelled 5 m. Step 2: a bracket over the whole bar with the words 26 m. Step 3: the words 3x + 5 = 26. Step 4: the words x = (26 − 5) ÷ 3 = 7.xxx5 m

    1 Let x be the length of one equal piece, in metres. Draw 3 pieces of x and the 5 m piece.

  2. Step 1: a bar with 3 equal shaded parts, each labelled x, and one part labelled 5 m. Step 2: a bracket over the whole bar with the words 26 m. Step 3: the words 3x + 5 = 26. Step 4: the words x = (26 − 5) ÷ 3 = 7.xxx5 m26 m

    2 The whole rope is 26 m. Write that over the bar.

  3. Step 1: a bar with 3 equal shaded parts, each labelled x, and one part labelled 5 m. Step 2: a bracket over the whole bar with the words 26 m. Step 3: the words 3x + 5 = 26. Step 4: the words x = (26 − 5) ÷ 3 = 7.xxx5 m26 m3x + 5 = 26

    3 Write the equation from the bar: 3x + 5 = 26.

  4. Step 1: a bar with 3 equal shaded parts, each labelled x, and one part labelled 5 m. Step 2: a bracket over the whole bar with the words 26 m. Step 3: the words 3x + 5 = 26. Step 4: the words x = (26 − 5) ÷ 3 = 7.xxx5 m26 m3x + 5 = 26x = (26 − 5) ÷ 3 = 7

    4 Solve it. Take 5 from 26 and divide by 3. This gives x = 7.

Worked example

Three pieces of rope of the same length and a 5 m piece are joined. The rope is 26 m long. How long is each of the 3 pieces?

  1. Let x be the length of one piece, in metres.
  2. Three pieces and 5 m make 26 m, so 3x + 5 = 26.
  3. Take 5 from both sides: 3x = 26 − 5 = 21.
  4. Divide both sides by 3: x = 21 ÷ 3 = 7.
  5. Check in the equation: 3 × 7 + 5 = 21 + 5 = 26. It is true.
  6. Check in the story: three pieces of 7 m and one of 5 m make 26 m. Each piece is 7 m.

Answer: 7 m

Formulas are equations too. If you know the perimeter and one side of a rectangle, you can find the other side.

A rectangle 4 cm wide. The top and the bottom are marked length l. The left side and the right side are marked 4 cm. Under it are the words perimeter 22 cm.length llength l4 cm4 cmperimeter 22 cm
The perimeter is 22 cm. We find the length l.

Worked example

A rectangle is 4 cm wide. Its perimeter is 22 cm. How long is it?

  1. The formula is P = 2l + 2w. Put in P = 22 and w = 4: 22 = 2l + 2 × 4.
  2. Work out the last part: 2 × 4 = 8.
  3. So 2l + 8 = 22. Take 8 from both sides: 2l = 22 − 8 = 14.
  4. Divide both sides by 2: l = 14 ÷ 2 = 7.
  5. Check: 2 × 7 + 2 × 4 = 14 + 8 = 22. It is true.

Answer: 7 cm

See it three ways

With things

Think of a number. Double it. Add 6. Tell a friend the answer, which is 20. Your friend must find your number.

As a picture

A number machine. The number n = 7 goes in. The first step is × 2 and gives 14. The second step is + 6 and gives 20.7n× 214+ 620

In symbols

2n + 6 = 20, so 2n = 14 and n = 7.

Try it

Think of a number

You need: a friend, paper and a pencil.

  1. Think of a number, and write it down where your friend cannot see it.
  2. Multiply it by 3, then add 4. Tell your friend only the answer.
  3. Your friend writes the equation 3n + 4 = your answer, and solves it.
  4. Show your number. Is it the same as the one your friend found?

What you will see: The equation always finds the number you thought of. If you thought of 5, the answer you told was 19, and 3n + 4 = 19 gives n = 5.

If you cannot try it, this is what happens.
A number machine. The number n = 5 goes in. The first step is × 3 and gives 15. The second step is + 4 and gives 19.5n× 315+ 419

A wrong idea many people have

If my answer fits my equation, my answer must be right.

Why it feels right: The numbers match, so everything seems fine.

The test that breaks it: Suppose you write 3x − 5 = 16 for the rope story by mistake. You get x = 7. Check: 3 × 7 − 5 = 16. It fits. But 3 pieces of 7 m and 5 m make 26 m, not 16 m.

What is true: Check in the story as well as in the equation. A wrong equation can still fit its own answer.

Key words

word problem
a story with numbers, to be turned into an equation.
solution
the number that makes an equation true.

Back to your guess

The answer is C: Each piece is 7 m. The equation 3x + 5 = 26 gives x = 7.

Let x be the length of one piece. Three pieces and 5 m make 26 m, so 3x + 5 = 26. Take 5 from both sides, then divide by 3. Each piece is 7 m.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Joseph has n beans. He gets 4 more and then has 15. Which equation is right, and what is n?

    Tap your answer.

    1. Not this one. A minus sign means he gave 4 away. Joseph gets 4 more, so we add. Go to M3.8.7. Try another.

    2. Yes, that is right. Joseph gets 4 more, so n + 4 = 15. Take 4 from both sides: n = 15 − 4 = 11. Check: 11 + 4 = 15.

    3. Not this one. 4n means 4 times n. Joseph does not have 4 boxes. He gets 4 more beans. Go to M3.8.1. Try another.

    See the answer

    n + 4 = 15, so n = 11

    Joseph gets 4 more, so n + 4 = 15. Take 4 from both sides: n = 15 − 4 = 11. Check: 11 + 4 = 15.

    • If you chose A: A minus sign means he gave 4 away. Joseph gets 4 more, so we add. Go to M3.8.7.
    • If you chose C: 4n means 4 times n. Joseph does not have 4 boxes. He gets 4 more beans. Go to M3.8.1.

  2. Amina thinks of a number. She doubles it and takes away 3. The answer is 11. Which equation is right, and what is her number?

    Tap your answer.

    1. Yes, that is right. Add 3 to both sides: 2n = 14. Divide by 2: n = 7. Check: 2 × 7 − 3 = 11.

    2. Not this one. You missed the doubling. The number is doubled first, so we write 2n. Go to M3.8.7. Try another.

    3. Not this one. She takes 3 away. A plus sign would add 3. Go to M3.8.7. Try another.

    See the answer

    2n − 3 = 11, so n = 7

    Add 3 to both sides: 2n = 14. Divide by 2: n = 7. Check: 2 × 7 − 3 = 11.

    • If you chose B: You missed the doubling. The number is doubled first, so we write 2n. Go to M3.8.7.
    • If you chose C: She takes 3 away. A plus sign would add 3. Go to M3.8.7.

  3. The bar shows 4 equal pieces of rope and one piece of 3 m. The whole rope is 19 m. How long is each equal piece?

    One bar with the words 19 m over it. It is cut into 4 equal shaded parts, each labelled x, and one part labelled 3 m.xxxx3 m19 m
    Tap your answer.

    1. Not this one. You took 3 from 19, which is right, but you stopped. 16 m is all 4 pieces. Divide by 4. Go to M3.8.7. Try another.

    2. Not this one. You added 3 to 19 and divided by 4. The 3 m is part of the rope, so take it away from 19. Go to M3.8.7. Try another.

    3. Yes, that is right. The equation is 4x + 3 = 19. Take 3 from both sides: 4x = 16. Divide by 4: x = 4. Check: 4 × 4 + 3 = 19.

    See the answer

    4 m

    The equation is 4x + 3 = 19. Take 3 from both sides: 4x = 16. Divide by 4: x = 4. Check: 4 × 4 + 3 = 19.

    • If you chose A: You took 3 from 19, which is right, but you stopped. 16 m is all 4 pieces. Divide by 4. Go to M3.8.7.
    • If you chose B: You added 3 to 19 and divided by 4. The 3 m is part of the rope, so take it away from 19. Go to M3.8.7.

  4. Lucía says the solution of 5x + 2 = 32 is x = 7. Put 7 back into the equation. What do you find?

    Tap your answer.

    1. Not this one. The right side is 32. A true answer must make the left side 32, not 37. Go to M3.8.7. Try another.

    2. Yes, that is right. The right answer is x = 6, because 5 × 6 + 2 = 32. The 7 does not fit, so Lucía made a slip.

    3. Not this one. 5x means 5 times x. Put 7 in place of x, then multiply: 5 × 7. Go to M3.8.2. Try another.

    See the answer

    5 × 7 + 2 = 37, not 32. She is wrong.

    The right answer is x = 6, because 5 × 6 + 2 = 32. The 7 does not fit, so Lucía made a slip.

    • If you chose A: The right side is 32. A true answer must make the left side 32, not 37. Go to M3.8.7.
    • If you chose C: 5x means 5 times x. Put 7 in place of x, then multiply: 5 × 7. Go to M3.8.2.

  5. A rectangle is 6 cm wide. Its perimeter is 30 cm. Which equation finds the length l?

    A rectangle 6 cm wide. The top is marked length l. The left side is marked 6 cm.length l6 cm
    Tap your answer.

    1. Not this one. This adds only one length and one width. The perimeter goes round all four sides. Go to M3.8.2. Try another.

    2. Not this one. You doubled the length but not the width. There are two widths, so 2 × 6. Go to M3.8.2. Try another.

    3. Not this one. This multiplies the sides, which is the area. The perimeter adds the four sides. Go to M3.8.2. Try another.

    4. Yes, that is right. P = 2l + 2w with w = 6 gives 2l + 2 × 6 = 30, so 2l + 12 = 30. Then 2l = 18 and l = 9.

    See the answer

    2l + 12 = 30

    P = 2l + 2w with w = 6 gives 2l + 2 × 6 = 30, so 2l + 12 = 30. Then 2l = 18 and l = 9.

    • If you chose A: This adds only one length and one width. The perimeter goes round all four sides. Go to M3.8.2.
    • If you chose B: You doubled the length but not the width. There are two widths, so 2 × 6. Go to M3.8.2.
    • If you chose C: This multiplies the sides, which is the area. The perimeter adds the four sides. Go to M3.8.2.

  6. From an earlier chapter

    Solve 3x + 4 = 19.

    Tap your answer.

    1. Yes, that is right. Take 4 from both sides: 3x = 15. Divide by 3: x = 5. Check: 3 × 5 + 4 = 19.

    2. Not this one. You took 4 from 19, which is right, but you stopped. This is 3x = 15. Divide by 3. Go to M3.8.6. Try another.

    3. Not this one. You added 4 to 19 and stopped. The 4 is added to 3x, so take it away. Go to M3.8.6. Try another.

    See the answer

    5

    Take 4 from both sides: 3x = 15. Divide by 3: x = 5. Check: 3 × 5 + 4 = 19.

    • If you chose B: You took 4 from 19, which is right, but you stopped. This is 3x = 15. Divide by 3. Go to M3.8.6.
    • If you chose C: You added 4 to 19 and stopped. The 4 is added to 3x, so take it away. Go to M3.8.6.

  7. I think of a number. I multiply it by 3 and add 4. The answer is 19. What is my number?

    See the answer

    5

    The equation is 3n + 4 = 19. Take 4 from both sides: 3n = 15. Divide by 3: n = 5. Check: 3 × 5 + 4 = 19.

  8. Joseph has some beans. He gives 7 away and has 12 left. How many beans did he have at the start?

    See the answer

    19 beans

    The equation is n − 7 = 12. Add 7 to both sides: n = 19. Check: 19 − 7 = 12.

Keep it

Copy this into your exercise book.

Pick a letter and say what it means. Write the equation from the story. Solve it. Check in the equation and in the story.

Draw: Draw a bar for the rope story. It has 3 parts of x and a part of 5. Write 3x + 5 = 26.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (850 KB). It opens on any phone.