Depth · one concept from first principles · step 2 of 4

Simplify the expression before comparing

Find every natural number that $\blacksquare$ can be.

$9\frac{1}{6} \div 2\frac{3}{4} < 42 \div \frac{7}{\blacksquare} < 28$

Number systemExpressions

Your answer

How to solve

Strategy Convert to Algebra — Nothing can be compared while the parts are still tangled up. Simplify the left side to a single number and the middle to a single multiple of the blank; only then does the condition read as a range.

1STEP 1

Simplify the left side

Write both mixed numbers as improper fractions and turn the division into a multiplication by the reciprocal.

55/6 × 4/11 = 10/3

2STEP 2

Simplify the middle

Dividing by 7/(■) means multiplying by ■/7, and 42 ÷ 7 = 6.

42 ÷ 7 = 6

3STEP 3

Turn the condition into a range for the blank

Divide all three parts by 6; dividing by a positive number keeps the signs pointing the same way.

10/3 ÷ 6 = 5/9, 28 ÷ 6 = 14/3

4STEP 4

List the natural numbers inside

5/9 is under 1 and 14/3 is between 4 and 5, so the natural numbers from 1 to 4 fit.

■ = 1, 2, 3, 4

Answer

1, 2, 3, 4

Test the ends: with ■ = 4 the middle is 24, which is under 28; with ■ = 5 it is 30, which is not.

Takeaway

Simplify each side to a single number before comparing -- the range only appears once the clutter is gone.

Simplify the left side
Simplify the middle
Turn the condition into a range for the blank
List the natural numbers inside

Where next?

Another one like thissuggested

12 practice problems

Step back and solidifysuggested

Overlap of two ranges on a number line Inclusive vs exclusive range boundaries

1 reasoning problems grow out of this type — the bridge toward competition →

A parent dashboard for the family lives at sensimlab.com.

▶ Practice — 12 problems

The same shape shows up later on the AMC, as Small-Domain Constrained Search. Why the shapes repeat →