Raven holds up two jars. "Rocket's jar has 24 beans. Mine has half again as many. How many is that?"
"Twenty-four plus half of 24," Rocket says. "24 + 12 = 36."
"Write it with a letter," Raven says. "What do you notice when the jar is a?"
"a + 0.5a," Rocket says. He stares at it. "That is just 1.5a. Half again as much means multiply by 1.5."
Nova hovers between the jars, her light bright. "Rewriting the expression showed you the puzzle," she says.
"And 1.5 × 24 is easy to estimate," Raven says. "A bit more than 24 plus 12. Thirty-something."
Rocket checks. "36. The estimate agrees." Nova hums. "Estimate first, then trust the exact answer."
An expression can be rewritten in a different form, and the new form can show what the puzzle is really saying.
a + 0.5a means a jar plus half a jar. Combine the like terms: a + 0.5a = 1.5a. Half again as much is multiply by 1.5.
In the same way, a - 0.25a = 0.75a: a quarter less is three quarters of the whole. And 2a + a = 3a: double plus one more is triple.
| The puzzle says | Expression | Rewritten | Which means |
|---|---|---|---|
| half again as much | a + 0.5a | 1.5a | multiply by 1.5 |
| a quarter less | a - 0.25a | 0.75a | multiply by 0.75 |
| double, plus one more | 2a + a | 3a | multiply by 3 |
| a tenth more | a + 0.1a | 1.1a | multiply by 1.1 |
| Jar | Beans | Half again as many |
|---|---|---|
| Rocket | 24 | 36 |
| Raven | 30 | 45 |
| Nova | 18 | 27 |
These three jars are the crew's own counts. Each "half again" column is 1.5 times the beans, the rewritten form at work.
With decimals and fractions, estimate first. Round to friendly numbers, work it out in your head, then compare with the exact answer.
For 1.9x + 3.2 = 12.7, think 2x + 3 = 13, so x is about 5. The exact answer is x = 5. The estimate agrees.
Rewritten and estimated. Tomorrow you find two-step equations in everyday life and review the week.