January 2026 Contest — Grade 7

Question 1

Two snails, Turbo and Speedy, start at opposite ends of a 10-meter path. Turbo travels 2/3 of the length of the path. Speedy travels 3/4 of the length of the path from the other end. What fraction of the path's length represents the overlapping section where both snails have traveled?

Accepted Answer

To find the fraction of the path's length where both snails have traveled (the overlapping section), we can represent the total length of the path as 1 (or one whole).

1. Turbo starts at one end and travels 2/3 of the path's length.
2. Speedy starts at the opposite end and travels 3/4 of the path's length.

Since they start at opposite ends and move toward each other, any distance they travel beyond the total length of the path represents the overlapping section.

We can find this overlap by adding their individual fractional distances and subtracting the total length of the path (1):
Overlap = 2/3 + 3/4 - 1

To add the fractions, we find a common denominator, which is 12:
2/3 = 8/12
3/4 = 9/12

Now, add the two fractions:
8/12 + 9/12 = 17/12

Finally, subtract the total length of the path (which is 1 = 12/12):
17/12 - 12/12 = 5/12

Thus, the overlapping section represents 5/12 of the path's length.

The answer is C

Question 2

The numbers 1 through 9 are entered into a 3x3 grid. The sums of the rows and columns are computed. Five of the sums are 12, 13, 15, 16, and 17. What is the sixth sum?

Accepted Answer

Step 1 — Read the image
The image shows a blank 3 x 3 grid containing 9 individual squares.

Step 2 — Solve
Let the numbers placed in the 3 x 3 grid be represented by the variables in the positions below:
begin{array}{|c|c|c|}
hline
a & b & c \
hline
d & e & f \
hline
g & h & i \
hline
end{array}

The numbers 1 through 9 are entered into the grid, meaning each number from \{1, 2, 3, 4, 5, 6, 7, 8, 9\} is used exactly once.

The sum of all numbers in the grid is:
S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45

Now, let's look at the row sums and column sums:
The three row sums are:
 R\_1 = a + b + c
 R\_2 = d + e + f
 R\_3 = g + h + i
 The sum of these three row sums is equal to the sum of all elements in the grid:
 R\_1 + R\_2 + R\_3 = S = 45

The three column sums are:
 C\_1 = a + d + g
 C\_2 = b + e + h
 C\_3 = c + f + i
 The sum of these three column sums is also equal to the sum of all elements in the grid:
 C\_1 + C\_2 + C\_3 = S = 45

Therefore, the total sum of all 6 sums (3 row sums and 3 column sums) is:
(R\_1 + R\_2 + R\_3) + (C\_1 + C\_2 + C\_3) = 45 + 45 = 90

We are given five of these sums: 12, 13, 15, 16, and 17. Let the sixth sum be x. We can write the equation:
12 + 13 + 15 + 16 + 17 + x = 90
73 + x = 90
x = 17

To verify that such a grid exists, we can partition the six sums \{12, 13, 15, 16, 17, 17\} into two sets of three that each sum to 45:
Row sums: \{12, 16, 17\} (since 12 + 16 + 17 = 45)
Column sums: \{13, 15, 17\} (since 13 + 15 + 17 = 45)

We can construct a valid grid using the numbers 1 through 9 exactly once:
begin{array}{|c|c|c|c}
hline
1 & 4 & 7 & Row Sum = 12 \
hline
3 & 5 & 8 & Row Sum = 16 \
hline
9 & 6 & 2 & Row Sum = 17 \
hline
end{array}
Col Sums: 13 quad 15 quad 17

This grid is valid, and the sixth sum is indeed 17.

Step 3 — Select
The correct option is E.

The answer is E

January 2026 Contest — Grade 7

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