Halves, Quarters, Thirds

More pieces means each piece is smaller.

The name comes from the cut

Cut a sandwich into 2 equal pieces, and each piece is called a half. Cut it into 4 equal pieces, and each piece is a quarter. Cut it into 3 equal pieces, and each piece is a third. The name of a piece says how many equal pieces the whole was cut into.

The pieces must be the same size. A sandwich cut into one big piece and one small piece is in 2 pieces, but neither piece is a half.

More pieces, smaller pieces

Share one pizza between 2 people and each gets a half. Share the same pizza between 4 people and each gets a quarter, which is smaller, because the pizza went further. A third sits in between: more pieces than halves, so each is smaller than a half, and fewer pieces than quarters, so each is bigger than a quarter.

The pieces also say how many of them make the whole: 2 halves, 3 thirds, 4 quarters.

11/21/31/41/8

Every row is the same whole, cut into more pieces going down. One half is longer than one third, one third is longer than one quarter, and one eighth is the shortest.

Writing one piece

One piece is written as 1 over the number of pieces. A half is 1/2, a third is 1/3 and a quarter is 1/4. Cut a whole into 8 equal pieces and each piece is an eighth, written 1/8.

The bottom number counts pieces; it is not a size. 8 is bigger than 3, but 1/8 is smaller than 1/3, because the whole was cut into more pieces. More cuts make smaller pieces.

1/8

1/8 = 0.125

Make one half

One piece stays shaded while the number of pieces changes. Fewer pieces make each piece bigger, from 1/8 up to 1/2.

Worked example: Pieces of a Play Mat Put Together to Make the Whole

Question A play mat is a square made of 4 equal pieces. Each piece is 14 of the mat. Jun has put down 1 piece. (a) How many more pieces does Jun need to make the whole mat? (b) Mei's mat is the same size. Each of her pieces is 12 of the mat. How many of her pieces make the whole mat?

  1. 1.Draw Jun's mat as a square with 4 equal parts. Shade the 1 piece he has put down.

    Jun's mat1/4
    Jun's mat1/4
    Jun's mat has 4 equal parts. The 1 piece he has put down is shaded.

  2. 2.Count on until the square is full: 1 quarter, 2 quarters, 3 quarters, 4 quarters. Four quarters make 1 whole.

    Jun's mat12344 quarters = 1 whole
    Jun's mat12344 quarters = 1 whole
    Count on until the square is full: 1 quarter, 2 quarters, 3 quarters, 4 quarters. Four quarters make 1 whole.

  3. 3.(a) Jun has 1 piece, so he needs 4 − 1 = 3 more pieces.

    Jun's mat12344 − 1 = 3 more
    Jun's mat12344 − 1 = 3 more
    (a) Jun has 1 piece, so he needs 4 − 1 = 3 more pieces.

  4. 4.Draw Mei's mat with 2 equal parts. Each part is 12 of the mat.

    Jun's mat12344 − 1 = 3 moreMei's mat1/21/2
    Jun's mat12344 − 1 = 3 moreMei's mat1/21/2
    Mei's mat has 2 equal parts. Each part is 12 of the mat.

  5. 5.(b) Count on until the square is full: 1 half, 2 halves. Two halves make 1 whole, so 2 of her pieces make the whole mat.

    Jun's mat12344 − 1 = 3 moreMei's mat1/21/22 halves = 1 whole
    Jun's mat12344 − 1 = 3 moreMei's mat1/21/22 halves = 1 whole
    (b) Count on: 1 half, 2 halves. Two halves make 1 whole, so 2 of her pieces make the whole mat.

Answer: (a) 3 more pieces; (b) 2 pieces

Common mistakes

  • Answering 4 for part (a). Four quarters make the whole mat, but Jun has already put down 1 of them, so he needs only 3 more.
  • Saying that Mei also needs 4 pieces because her mat is the same size. Her pieces are bigger. One half covers as much as 2 quarters, so only 2 halves fill the mat.

More halves and quarters problems, worked step by step →

Worked example: A Quarter of an Hour and Half an Hour in Minutes

Question There are 60 minutes in 1 hour. Tim reads for a quarter of an hour. Then he plays for half an hour. (a) For how many minutes does Tim read? (b) For how many minutes does Tim play?

  1. 1.Draw 1 hour as a bar of 12 blocks. Each block is 5 minutes. Count in fives along the bar to check: the last block ends at 60 minutes.

    1 hour55555555555560 minutes
    1 hour55555555555560 minutes
    One hour is a bar of 12 blocks. Each block is 5 minutes, and the last block ends at 60 minutes.

  2. 2.A quarter is 1 of 4 equal parts. Cut the bar into 4 equal parts. Each part has 3 blocks.

    quarters55555555555560 minutes3 blocks3 blocks3 blocks3 blocks
    quarters55555555555560 minutes3 blocks3 blocks3 blocks3 blocks
    A quarter is 1 of 4 equal parts. Each part of the bar has 3 blocks.

  3. 3.(a) Count in fives along 3 blocks: 5, 10, 15. A quarter of an hour is 15 minutes, so Tim reads for 15 minutes.

    quarters5101560 minutes15 minutes1/4 hour = 15 minutes
    quarters5101560 minutes15 minutes1/4 hour = 15 minutes
    (a) Count in fives along 3 blocks: 5, 10, 15. A quarter of an hour is 15 minutes.

  4. 4.Half is 1 of 2 equal parts. Cut the bar into 2 equal parts. Each part has 6 blocks.

    quarters5101560 minutes15 minuteshalves5555555555556 blocks6 blocks1/4 hour = 15 minutes
    quarters5101560 minutes15 minuteshalves5555555555556 blocks6 blocks1/4 hour = 15 minutes
    Half is 1 of 2 equal parts. Each part of the bar has 6 blocks.

  5. 5.(b) Count in fives along 6 blocks: 5, 10, 15, 20, 25, 30. Half an hour is 30 minutes, so Tim plays for 30 minutes. Check: 30 + 30 = 60.

    quarters5101560 minutes15 minuteshalves5101520253030 minutes1/4 hour = 15 minutes1/2 hour = 30 minutes
    quarters5101560 minutes15 minuteshalves5101520253030 minutes1/4 hour = 15 minutes1/2 hour = 30 minutes
    (b) Count in fives along 6 blocks: 5, 10, 15, 20, 25, 30. Half an hour is 30 minutes.

Answer: (a) 15 minutes; (b) 30 minutes

Common mistakes

  • Saying that a quarter of an hour is 25 minutes. That would be right if an hour had 100 minutes. An hour has 60 minutes, and 60 minutes cut into 4 equal parts gives 15 minutes in each part.
  • Writing 4 minutes for a quarter of an hour because of the 4 in 14. The 4 is the number of equal parts, not the number of minutes in a part.

More halves and quarters problems, worked step by step →

Practice Halves, Quarters, Thirds in the app