Fraction Claims: True or False on a Number Line

Three people share a chocolate bar: Andy eats a fifth, Bella two fifths, and Carlo the rest. First find the rest: one minus a fifth minus two fifths is two fifths. Then put everything on one number line from zero to one, including the benchmarks a quarter, a half and three quarters. Choose a denominator all the fractions fit into — fifths and quarters both go into twentieths — and every fraction becomes a whole number of twentieths. Now any claim is a glance along the line. Two fifths is left of a half, so Carlo ate less than half. A fifth is left of a quarter, so Andy did not eat more than a quarter.

Test claims are almost always measured against a quarter, a half and three quarters, and each has a quick test. More than a half: double the top and see if it beats the bottom — three eighths doubled is six, less than eight, so under a half. More than a quarter: multiply the top by four — a fifth gives four, less than five, so under a quarter. More than three quarters: four times the top against three times the bottom — four fifths gives sixteen against fifteen, so over. With those three tests the number line is a check rather than a necessity.

A claim about two people together is a claim about their sum, so add first. Andy's fifth and Bella's two fifths make three fifths. Three fifths against three quarters: twelve twentieths against fifteen, so less — the claim is true. Rio's three eighths and Sam's quarter make five eighths, which is over a half and under three quarters. Add with matching pieces, then place the total on the line and test it. The mistake is to test the two people separately and then somehow combine the answers.

Now the whole question. Three statements about Andy, Bella and Carlo. Take them one at a time. Carlo ate less than half: two fifths is eight twentieths, a half is ten, so yes. Andy ate more than a quarter: a fifth is four twentieths, a quarter is five, so no. Andy and Bella ate less than three quarters altogether: three fifths is twelve twentieths, three quarters is fifteen, so yes. Statements one and three. Tick or cross every statement before you look at the options; the options are built to catch anyone who stops after the first tick.

The routine, so it is automatic on the day. Find any missing share as one minus the others. Choose a common denominator that takes every fraction and the benchmarks. Rename them all. For altogether, add; for how much more, subtract. Then tick or cross each statement and pick the option that matches. The marks are lost in three places: forgetting to work out the rest, testing people separately when the claim is about them together, and thinking a fifth is bigger than a quarter because five is bigger than four. Quiz and exercises now.