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, 2014

Algebra Difficulty 2.5 Junior Find the answer Canada

In the $4\times 4$ grid shown, each of the four symbols has a different value. The sum of the values of the symbols in each row is given to the right of that row.

Hide/Reveal Description of Grid

Each square in the 4 by 4 grid has a heart, a triangle, a square, or a diamond. The shapes in each row, from left to right, and the number to the right of the row is given in the following list:

Row 1 has heart, triangle, triangle, heart and the number 26.
Row 2 has four triangles and the number 24.
Row 3 has square, diamond, heart, diamond and the number 27.
Row 4 has square, heart, square, triangle and the number 33.

What is the value of $\blacklozenge$ ?

Pick one

Solution

From the second row, $\triangle + \triangle + \triangle + \triangle = 24$ or $4\triangle = 24$ , and so $\triangle = 6$ .

From the first row, $\heartsuit + \triangle + \triangle + \heartsuit = 26$ or $2 \heartsuit + 2 \triangle = 26$ .

Since $\triangle = 6$ , then $2 \heartsuit = 26 - 12 = 14$ , and so $\heartsuit = 7$ .

From the fourth row, $\square + \heartsuit + \square + \triangle = 33$ .

Since $\triangle = 6$ and $\heartsuit = 7$ , then $2\square + 7 + 6 = 33$ , and so $2\square = 20$ or $\square = 10$ .

Finally, from the third row, $\square +$ $\blacklozenge +$ $\heartsuit +$ $\blacklozenge$ $=27$ .

Since $\square = 10$ and $\heartsuit = 7$ , then $2$ $\blacklozenge$ $= 27 - 10 - 7 = 10$ .

Thus, $\blacklozenge = 5$ .

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.