Question 11

We will assign variables to each shape and solve the resulting system of linear equations to find their values.

Step 1 — Find values for shapes in the first grid

Let us represent the blue square as 'S' and the red circle as 'C'. We can write equations from the given row totals.

From the first row, we have: $S + S + C = 27$ $2S + C = 27 \quad \text{(Equation 1)}$

From the second row, we have: $C + C + S = 21$ $S + 2C = 21 \quad \text{(Equation 2)}$

Now, we solve these two equations. Let us subtract Equation 2 from Equation 1. $(2S + C) - (S + 2C) = 27 - 21$ $S - C = 6$ $S = C + 6 \quad \text{(Equation 3)}$

Substitute the value of S from Equation 3 into Equation 2. $(C + 6) + 2C = 21$ $3C + 6 = 21$ $3C = 21 - 6$ $3C = 15$ $C = \frac{15}{3}$

$\boxed{C = 5}$

Now, substitute the value of C back into Equation 3 to find S. $S = 5 + 6$

$\boxed{S = 11}$

So, the value of a red circle is 5 and a blue square is 11.

Step 2 — Calculate the missing row total in the first grid

The third row has a red circle, a blue square, and a red circle. We use the values we found for C and S. $C + S + C$ $= 5 + 11 + 5$

$\boxed{21}$

Step 3 — Find values for shapes in the second grid

Let us represent the blue circle as 'BC' and the purple diamond as 'PD'. We can write equations from the given row totals.

From the first row, we have: $BC + PD + PD = 18$ $BC + 2PD = 18 \quad \text{(Equation 4)}$

From the second row, we have: $PD + BC + BC = 15$ $2BC + PD = 15 \quad \text{(Equation 5)}$

Now, we solve these two equations. Let us multiply Equation 5 by 2. $2 \times (2BC + PD) = 2 \times 15$ $4BC + 2PD = 30 \quad \text{(Equation 6)}$

Now, subtract Equation 4 from Equation 6. $(4BC + 2PD) - (BC + 2PD) = 30 - 18$ $3BC = 12$ $BC = \frac{12}{3}$

$\boxed{BC = 4}$

Now, substitute the value of BC back into Equation 5 to find PD. $2(4) + PD = 15$ $8 + PD = 15$ $PD = 15 - 8$

$\boxed{PD = 7}$

So, the value of a blue circle is 4 and a purple diamond is 7.

Step 4 — Calculate the missing row total in the second grid

The third row has a purple diamond, a blue circle, and a blue circle. We use the values we found for PD and BC. $PD + BC + BC$ $= 7 + 4 + 4$

$\boxed{15}$

Step 5 — Calculate the missing column totals in the second grid

Let us find the total for each column. For the first column: $BC + PD + PD$ $= 4 + 7 + 7$

$\boxed{18}$

For the second column: $PD + BC + BC$ $= 7 + 4 + 4$

$\boxed{15}$

For the third column: $PD + BC + BC$ $= 7 + 4 + 4$

$\boxed{15}$

Answer

(i) Value of a blue square: 11 (ii) Value of a red circle: 5 (iii) Value of a blue circle: 4 (iv) Value of a purple diamond: 7 (v) Missing row total in the first grid: 21 (vi) Missing row total in the second grid: 15 (vii) Missing column 1 total in the second grid: 18 (viii) Missing column 2 total in the second grid: 15 (ix) Missing column 3 total in the second grid: 15