Problem
Bound an inequality at equality
We need every digit from 1 to 9 that can fill the box so that 9 times the box is larger than 6 times 8.
Operations
Your answer
Watch out The problem asks for every number that works, so stopping at the smallest box that first beats 48 leaves the rest out.
How to solve
Strategy Guess and Check — First compute the fixed left side, then test the box values in order from small to large to find where 9 x box first passes 48; listing the candidates keeps every value from 1 to 9 accounted for.
1STEP 1
Compute the left side
Work out 6 times 8 so the inequality becomes a comparison with a single number.
6 × 8 = 48
6 times 8 is a basic multiplication fact, so the left side is just the number 48.
3.OA.C.7Guess And Check2STEP 2
Find where 9 times the box passes 48
Test multiples of 9 in order: 9x5 = 45 is not greater than 48, but 9x6 = 54 is greater than 48. So the box must be 6 or larger.
9 × 5 = 45, 9 × 6 = 54
Checking the nine-times facts shows exactly when the right side becomes bigger than 48.
3.OA.A.4Guess And Check9 times the first rises above 48 at = 6, so the has to be 6 or larger.
Why?
A of 5 makes the right side 9 x 5 = 45, which is still under 48, so a of 5 or anything smaller is too small.
Why?
9 x 5 is nine equal groups of 5, and nine groups of 5 pile up to only 45.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
A of 6 makes the right side 9 x 6 = 54, and 54 is above 48, so a of 6 already works.
Why?
9 x 6 is nine equal groups of 6, and nine groups of 6 pile up to 54.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
Raising the from 5 to 6 adds one more to each of the nine groups, so the total climbs from 45 by another 9 up to 54, past 48.
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
Why?
A bigger than 6 puts even more into each of the nine groups, so the right side only grows and stays above 48 -- the just needs to be 6 or more.
Why?
Every time the goes up by one, all nine groups each gain one, so the total gains another 9 and can never shrink.
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
3STEP 3
List all box values that work
Every box value 6 and above (and at most 9) makes the right side larger than 48, so collect them.
□ ∈ {6, 7, 8, 9}
Since bigger box values only make 9 x box larger, all of 6, 7, 8, 9 keep the inequality true.
3.OA.A.4Make A Systematic ListAnswer
6, 7, 8, 9
Checking the smallest answer: 9x6 = 54 ＞ 48, true; and the value just below, 9x5 = 45, is not greater than 48, so 6 is correctly the cutoff and 6, 7, 8, 9 all work.
Takeaway
Turn one side into a number, then test the times-table facts -- this is just Grade 3 multiplication you already know!
- Compute the left side
- Find where 9 times the box passes 48
- List all box values that work
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested