Finding the whole from a fraction

In a box, one quarter of the beads are red and 3 eighths of the beads are blue. The remaining 30 beads are yellow. How many beads are there in the box? Try it first. Pause if you like. Which fraction of the whole does the number 30 measure? Draw one bar for all the beads in the box. Cut the bar into 4 equal parts. One quarter of the beads are red. Cut each part into 2 equal parts. Now there are 8 equal parts. 3 eighths of the beads are blue. The remaining 30 beads are yellow. The fraction of yellow beads is one minus one quarter minus 3 eighths. Rename the whole as 8 eighths and one quarter as 2 eighths. 8 eighths minus 2 eighths minus 3 eighths equals 3 eighths. 3 eighths of the beads are yellow. Let the number of beads be 8 units. Each equal section of the bar is one unit. 3 units equal 30. The 30 yellow beads belong to these 3 units together. One unit equals 30 divided by 3, which is 10. The question asks for every bead. How many equal units make the whole? 8 units equal 8 times 10, which is 80. There are 80 beads in the box. Find the fraction that the known count measures. Use that count to find one unit, then multiply to find the whole.

One half of the pupils in a hall chose swimming, one third chose football and the rest chose badminton. 28 more pupils chose swimming than badminton. How many pupils were there in the hall? Try it first. Pause if you like. Does 28 measure one group, or the difference between two groups? Draw one bar for all the pupils in the hall. One half of the pupils chose swimming. Shade half of the bar. Divide the same bar into three equal parts. Each part is one third. Split each third into two equal parts. Now there are six equal parts. One third of the pupils chose football. One third is two of the six parts. The rest chose badminton. The fraction of pupils who chose badminton is one minus one half minus one third. Rename these fractions in sixths. 6 sixths minus 3 sixths minus 2 sixths equals one sixth. One sixth of the pupils chose badminton. Let the number of pupils be 6 units. Swimming equals 3 units and badminton equals one unit. Move the badminton unit beside swimming to compare the lengths. 3 units minus one unit equals 2 units. This is the extra length of the swimming bar. 2 units equal 28. The difference of 28 pupils measures the gap. One unit equals 28 divided by 2, which is 14. 6 units equal 6 times 14, which is 84. There were 84 pupils in the hall. Name what the given count measures. Find one unit, then the whole.