Math Detours · Grade 5 · Sharpen · 5

The same fraction in disguise

\(\tfrac12\), \(\tfrac24\) and \(\tfrac{50}{100}\) are one number with three names. Cut the pieces smaller and nothing changes but the label.

Name ____________________Date ____________

Tonight's idea

Shade half of a bar. Now cut every piece in two. There are 4 pieces and 2 are shaded. Cut again: 8 pieces, 4 shaded.

121 of 2 242 of 4 484 of 8
Cut the pieces smaller. The shaded part stops at the same dashed line every time.

The shaded part never moved. So \(\tfrac12\), \(\tfrac24\) and \(\tfrac48\) are the same number. Fractions like these are called equivalent, which means "equal in value".

Each cut doubled the number of pieces. That is the bottom number. It also doubled the number of shaded pieces. That is the top number.

Multiply the top and the bottom by the same number (not zero) and the fraction keeps its value.

It need not be doubling. Cut each piece into 3, or 5, or 50.

The rule runs backwards too. Divide the top and the bottom by the same number and the small pieces join up again. This is called simplifying. When no number except 1 divides both, the fraction is in its simplest form.

Watch one: simplify \(\tfrac{18}{24}\)

18 and 24 are both even. Divide both by 2: \(\tfrac{18}{24} = \tfrac{9}{12}\).

9 and 12 are both in the 3 times table. Divide both by 3: \(\tfrac{9}{12} = \tfrac34\). Only 1 divides both 3 and 4, so stop.

Now go the other way. Write \(\tfrac34\) as hundredths. \(4 \times 25 = 100\), so multiply the top by 25 as well: \(\tfrac34 = \tfrac{75}{100}\). You may know that one as 0.75.

Cut two bars smaller and watch the names change: mathdetours.com/demos/fractionbars.html.

Your turn

  1. Find the missing number: (a) \(\tfrac13 = \tfrac{\square}{12}\); (b) \(\tfrac25 = \tfrac{\square}{20}\); (c) \(\tfrac34 = \tfrac{15}{\square}\).
  2. Write each fraction in its simplest form: (a) \(\tfrac68\); (b) \(\tfrac{10}{15}\); (c) \(\tfrac{12}{36}\).
  3. Which is bigger, \(\tfrac23\) or \(\tfrac35\)? Rename both as fifteenths, then compare.
  4. Mia adds 1 to the top and the bottom of \(\tfrac12\) and gets \(\tfrac23\). "I did the same thing to both, so they are equal," she says. Draw two bars to show she is wrong. Which fraction is bigger?

Stretch: a disguise that adds to 100

A fraction is equal to \(\tfrac23\). Its top and bottom numbers add up to 100. What is the fraction?

Answers

  1. (a) 4, because \(3 \times 4 = 12\) and \(1 \times 4 = 4\). (b) 8. (c) 20, because the top was multiplied by 5.
  2. (a) \(\tfrac34\), dividing by 2. (b) \(\tfrac23\), dividing by 5. (c) \(\tfrac13\), dividing by 12.
  3. \(\tfrac23 = \tfrac{10}{15}\) and \(\tfrac35 = \tfrac{9}{15}\). Ten fifteenths beats nine, so \(\tfrac23\) is bigger.
  4. A bar cut in 2 with 1 shaded is half full. A bar cut in 3 with 2 shaded is more than half full, so \(\tfrac23\) is bigger. In sixths: \(\tfrac12 = \tfrac36\) and \(\tfrac23 = \tfrac46\). Multiplying both keeps the value. Adding to both does not.

Stretch. \(\tfrac{40}{60}\). In \(\tfrac23\) the top and bottom add to 5. To reach 100 you need 20 times as much, so multiply both by 20. Check: \(40 + 60 = 100\).

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