The votes are counted, the assembly is waiting—and three results have been written in different number bases. Decode them before the head teacher reaches the microphone.
YOUR MISSION PROGRESS • 0 OF 15 PRACTICE QUESTIONS SOLVED0%
The situation
Three teams. Three codes. One assembly.
At Kijani Secondary School, the mathematics club helped count votes for the student council election. To protect the results while they crossed the busy courtyard, each counting team used a different number base.
Then the results officer disappeared into a staff meeting—and forgot to leave the key! The assembly begins in one hour.
⚠ THE PROBLEM YOU MUST SOLVE
The three vote totals cannot be compared yet.
Team A wrote 101102, Team B wrote 435, and Team C wrote 278. The slips use different grouping rules, so the biggest-looking number may not be the biggest total.
Your job: change all three results into base ten, compare them fairly, and tell the assembly who won—or whether there is a tie.
Before the code • Begin with grouping
What does “base” actually mean?
A base is a grouping rule. It tells us how many ones must collect before we exchange them for one item in the next place-value column.
1310
You already use base ten every day.
When we collect 10 ones, we exchange them for 1 ten. So thirteen means:
1 group of ten + 3 loose ones.
That is why we write the digits 1 and 3: the 1 sits in the tens place and the 3 sits in the ones place.
TRY THE GROUPING RULE
We have 13 ballot papers.
Change the grouping rule and watch how the same amount is written differently.
✓✓✓✓✓✓✓✓✓✓✓✓✓
FULL BUNDLES11 group of 10
+
LOOSE ONES33 ones left
=
HOW WE WRITE IT13101 × 10 + 3 = 13
The quantity did not change. We still have 13 ballots. Only the grouping rule—and therefore the way we write the amount—changed.
THE BASE TELLS US
When to bundle
Base 5 bundles every five. Base 2 bundles every two. Base 8 bundles every eight.
THE DIGITS TELL US
How many bundles
In 235, the 2 means two bundles of five and the 3 means three loose ones.
THE SUBSCRIPT TELLS US
Which rule is being used
The small 5 in 235 is a label. It is not multiplied by 23.
Clue 1 • Read the base
The grouping rule also controls the digits.
Our everyday base-ten system uses digits 0 to 9. As soon as ten ones collect, we bundle them and move left—so we never need a single digit for “ten.” The same rule works in every base.
BASE 2
0, 1
Only two digits. A 2 can never appear in a base 2 number.
BASE 5
0, 1, 2, 3, 4
The largest allowed digit is always one less than the base.
BASE 8
0 to 7
Digits 8 and 9 cannot appear in a base 8 number.
QUICK CHECK 1
Which digit cannot appear in a base 5 number?
Clue 2 • Build the place-value house
Each step left is one larger bundle.
In base ten, each place is ten times the place to its right. In base five, each place is five times the place to its right. The places grow from the grouping rule.
FAMILIAR: BASE TEN
24310
HUNDREDS2 × 100TENS4 × 10ONES3 × 1
200 + 40 + 3 = 243
NEW: BASE FIVE
2435
GROUPS OF 252 × 25GROUPS OF 54 × 5ONES3 × 1
50 + 20 + 3 = 7310
Why 25? In base five, five groups of five make one larger bundle: 5 × 5 = 25. That is the place to the left of the fives column.
EXAMPLE A
10112
1 × 8+0 × 4+1 × 2+1 × 1= 1110
QUICK CHECK 2
What is 1011₂ in base ten?
EXAMPLE B
235
2 × 5+3 × 1= 1310
QUICK CHECK 3
What is 23₅ in base ten?
Clue 3 • Travel the other way
Convert base ten using repeated division.
To change 1910 to base 2, divide repeatedly by 2 and record each remainder. Read the remainders from bottom to top.
19 ÷ 2= 9 remainder 1
9 ÷ 2= 4 remainder 1
4 ÷ 2= 2 remainder 0
2 ÷ 2= 1 remainder 0
1 ÷ 2= 0 remainder 1
Read upward: 1910 = 100112
Practice studio • Questions 4–15
Now make the method yours.
Work one question at a time. If an answer is wrong, use the hint, repair the method and try again. Your solved-question count is saved on this device.
QUICK CHECK 4
Which digits are allowed in base 3?
QUICK CHECK 5
What are the place values in the three-digit base 4 number 231₄?
QUICK CHECK 6
Convert 31₄ to base ten.
QUICK CHECK 7
Convert 1101₂ to base ten.
QUICK CHECK 8
Convert 47₈ to base ten.
QUICK CHECK 9
Which base 5 number equals 19₁₀?
QUICK CHECK 10
Which binary number equals 10₁₀?
QUICK CHECK 11
What is 12₅ + 3₅, written in base 5?
QUICK CHECK 12
What is 14₅ + 2₅, written in base 5?
QUICK CHECK 13
Which statement is true?
QUICK CHECK 14
A number is written 68₈. What is wrong?
QUICK CHECK 15
Team D recorded 111₂ votes. How many votes is that?
Final scenario challenge
The head teacher has arrived.
Decode all three slips. Which announcement is mathematically correct?
TEAM A101102
TEAM B435
TEAM C278
Explain your thinking
Could the school simply announce Team C has 27 votes?
Write two or three sentences explaining why the digits alone do not tell us a number’s value. Use one result from the election as evidence.
Take the mission offline
Workbook and teaching support
Continue on paper, use the lesson in class, or send the activity home.